Multijet Cross Section Ratios

in p�p Collisions atps = 1:8 TeV

by

Ki Suk Hahn

Submitted in Partial Ful�llment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor Frederick Lobkowicz

and

Professor Gerald Blazey

Department of Physics and Astronomy

The College

Arts and Sciences

University of Rochester

Rochester, New York

1998

ii

To my parents.

iii

Curriculum Vitae

The author was born in Manila, Philippines on January 19, 1968. He attended

the California State Polytechnic University in Pomona from 1986 to 1988 and the

University of California, Los Angeles from 1988 to 1990, and graduated with a

Bachelor of Science degree in 1990. He came to the University of Rochester in the

Fall of 1991 for graduate studies in Physics. He received the Master of Arts degree

in 1993 and pursued his research in Particle Physics under the direction of the late

Professor Frederick Lobkowicz and Professor Gerald Blazey.

iv

Acknowledgments

I would like to thank the Rochester D� group for their friendship and help:

Kathy Fatyga, Eunil Won, Dylan Casey, Bob Hirosky, Stefan Gr�unendahl, Hui

Zhu, Cathy Cretsinger, Marc Paterno, Jae Yu, and professors Tom Ferbel and the

late Fred Lobkowicz.

For the analysis I thank Elizabeth Gallas and Jerry Blazey for their guidance,

and Soon Yung Jun for his friendship and help. I also thank David Summers,

Dieter Zeppenfeld, andWalter Giele for their comments and for patiently answering

questions about JETRAD.

I thank my uncles and aunts in NYC for their hospitality while I took much-wanted

breaks from schoolwork.

Additional thanks go to the sta�s at Fermilab and collaborating institutions for

their contributions to this work, and acknowledge support from the Department of En-

ergy and National Science Foundation (U.S.A.), Commissariat �a L'Energie Atomique

(France), State Committee for Science and Technology and Ministry for Atomic Energy

(Russia), CAPES and CNPq (Brazil), Departments of Atomic Energy and Science and

Education (India), Colciencias (Colombia), CONACyT (Mexico), Ministry of Education

and KOSEF (Korea), and CONICET and UBACyT (Argentina).

v

Abstract

We have measured the ratio of the three-jet to two-jet inclusive cross section as a

function of the total transverse energy of jets (HT =PEjT ) in events in pp collisions

atps = 1:8 TeV. We �nd that for

PET > 200 GeV there is a 70% probability

of emitting an additional jet with ET > 20 GeV. Using this measurement and the

framework of the theory of Quantum Chromodynamics (QCD), we investigated the size

of the renormalization scale used to model emission of \soft" (low-ET ) jets in multijet

�nal states. The �ndings indicate that, to order �3Sin QCD the data favor a scale

representative of the hard scattering rather than a softer scale of the order of the third

jet's transverse momentum. The preferred scale is �R � 14HT .

vi

Contents

List of Tables ix

List of Figures xiii

1 Introduction 1

1.1 Organization of thesis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3

2 QCD 4

2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4

2.2 Beginnings of QCD : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4

2.3 The Standard Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5

2.4 QCD : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7

2.5 Perturbative QCD : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8

2.6 Ratio of 3-jet and 2-jet cross sections : : : : : : : : : : : : : : : : : : : : : 16

3 The Tevatron and the D� Detector 17

3.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17

3.2 The Tevatron : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17

3.3 The D� detector : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20

3.3.1 The D� coordinate system : : : : : : : : : : : : : : : : : : : : : : 20

3.3.2 Central detector : : : : : : : : : : : : : : : : : : : : : : : : : : : : 22

3.3.3 Calorimeter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25

3.3.4 Muon system : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29

vii

4 Data Acquisition, Trigger, and O�ine Event Reconstruction 30

4.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30

4.1.1 Level 0 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31

4.1.2 Level 1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 32

4.1.3 Level 2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 33

4.2 O�ine processing : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 34

4.2.1 Farm : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 34

4.2.2 Event reconstruction : : : : : : : : : : : : : : : : : : : : : : : : : : 35

4.2.3 Data structure : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 38

5 The Data and Measurement of the Cross Section Ratio 40

5.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 40

5.2 Data sample : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 40

5.3 Event and jet selection criteria : : : : : : : : : : : : : : : : : : : : : : : : 42

5.3.1 Criteria for missing ET : : : : : : : : : : : : : : : : : : : : : : : : 42

5.3.2 Jet selection criteria : : : : : : : : : : : : : : : : : : : : : : : : : : 43

5.4 E�ciencies of the inclusive jet triggers : : : : : : : : : : : : : : : : : : : : 46

5.5 Measurement of the ratio of cross sections : : : : : : : : : : : : : : : : : 49

5.6 Minimum jet-ET requirement : : : : : : : : : : : : : : : : : : : : : : : : : 53

6 Sources of Measurement Uncertainty 56

6.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 56

6.2 Trigger e�ciency correction : : : : : : : : : : : : : : : : : : : : : : : : : : 56

6.3 Jet and event selection criteria : : : : : : : : : : : : : : : : : : : : : : : : 57

6.4 Vertex dependence : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 58

6.5 Luminosity dependence : : : : : : : : : : : : : : : : : : : : : : : : : : : : 60

6.6 Jet energy scale : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 61

6.7 Error summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 65

6.7.1 Statistical errors : : : : : : : : : : : : : : : : : : : : : : : : : : : : 65

6.7.2 Total systematic error : : : : : : : : : : : : : : : : : : : : : : : : : 67

viii

6.7.3 Correlated errors : : : : : : : : : : : : : : : : : : : : : : : : : : : : 67

7 Monte Carlo Simulations of QCD Processes 70

7.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 70

7.2 Monte Carlo prescription : : : : : : : : : : : : : : : : : : : : : : : : : : : 71

7.3 JETRAD : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 71

7.3.1 General approach : : : : : : : : : : : : : : : : : : : : : : : : : : : : 71

7.3.2 Input parameters and selection criteria : : : : : : : : : : : : : : : : 74

7.3.3 Sources of theoretical uncertainty : : : : : : : : : : : : : : : : : : 79

7.3.4 Renormalization scale of third-jet production : : : : : : : : : : : : 86

7.4 Comparison of data with QCD : : : : : : : : : : : : : : : : : : : : : : : : 88

7.4.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 88

7.4.2 Restricted range of jet pseudorapidity : : : : : : : : : : : : : : : : 94

8 Conclusion 96

Bibliography and Notes 98

A Matrix Elements for 2-to-3 Parton Scattering 103

B Calculation of �2 Using a Covariance Matrix 105

ix

List of Tables

2.1 Classi�cation of matter. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6

2.2 List of quarks. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7

2.3 Invariant matrix elements for 2-to-2 scattering. : : : : : : : : : : : : : : : 12

3.1 Tevatron and associated accelerators : : : : : : : : : : : : : : : : : : : : : 19

3.2 Central Detector Parameters : : : : : : : : : : : : : : : : : : : : : : : : : 24

3.3 Calorimeter Parameters : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28

5.1 Jet multiplicity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 41

5.2 Event and jet selection criteria : : : : : : : : : : : : : : : : : : : : : : : : 44

5.3 HT range used in each trigger. : : : : : : : : : : : : : : : : : : : : : : : : 51

6.1 Errors due to di�erent sources : : : : : : : : : : : : : : : : : : : : : : : : : 67

7.1 The scales for the hard scatter and third jet emission for two JETRAD

settings. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87

B.1 Data for sample �2 calculation. : : : : : : : : : : : : : : : : : : : : : : : 105

x

List of Figures

1.1 Two-jet and three-jet events. : : : : : : : : : : : : : : : : : : : : : : : : : 2

2.1 QCD description of a hard scattering process. : : : : : : : : : : : : : : : 9

2.2 Leading Order (�2S) diagrams for two-body scattering. : : : : : : : : : : : 11

2.3 Diagrams for Next-to-leading order (�3S) two-body scattering. : : : : : : 13

2.4 Pseudorapidity. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15

2.5 Higgs production via weak boson scattering. : : : : : : : : : : : : : : : : 16

3.1 The Fermilab Collider : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18

3.2 The D� Detector : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21

3.3 The Central Detector : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 22

3.4 The layout of sense and grid wires in the Vertex Drift Chambers. : : : : 23

3.5 A cross-sectional view of the Central Drift Chamber. : : : : : : : : : : : 24

3.6 An expanded view of the Forward Drift Chambers. : : : : : : : : : : : : 25

3.7 The D� calorimeter showing di�erent segmentation. : : : : : : : : : : : : 26

3.8 A schematic of a calorimeter cell. : : : : : : : : : : : : : : : : : : : : : : 26

3.9 Calorimeter segmentation : : : : : : : : : : : : : : : : : : : : : : : : : : : 27

4.1 D� trigger : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30

4.2 Level 0 Detectors : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31

4.3 The Level 1 Trigger System. : : : : : : : : : : : : : : : : : : : : : : : : : 32

4.4 O�ine Data Processing Farm : : : : : : : : : : : : : : : : : : : : : : : : : 34

xi

4.5 Preclustering. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 37

5.1 The distribution of jet multiplicities. : : : : : : : : : : : : : : : : : : : : 41

5.2 Logarithmic and linear plots of missing ET /E(1)T . : : : : : : : : : : : : : 43

5.3 Electromagnetic Fraction for all candidate jets. : : : : : : : : : : : : : : 44

5.4 Coarse hadronic fraction for all candidate jets. : : : : : : : : : : : : : : : 45

5.5 Hot cell ratio distribution. : : : : : : : : : : : : : : : : : : : : : : : : : : 46

5.6 Event trigger e�ciency vs HT for di�erent jet multiplicities (20 & 30 GeV

triggers). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 47

5.7 Event trigger e�ciency vs HT for di�erent jet multiplicities (50 & 85 GeV

triggers). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 48

5.8 Event trigger e�ciency vs HT for di�erent jet multiplicities (115 GeV

trigger). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 48

5.9 �3+�2+

for all �ve triggers. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 50

5.10 �3+�2+

for all �ve pseudo-triggers, using Monte Carlo : : : : : : : : : : : : : 52

5.11 �3+�2+

showing the breakdown of data points from di�erent triggers. : : : : 54

5.12 �3+�2+

for two values of mininum jet ET . : : : : : : : : : : : : : : : : : : : 55

6.1 Systematic error due to event and jet selection criteria. : : : : : : : : : : 57

6.2 Change in jet ET after re-vertexing. : : : : : : : : : : : : : : : : : : : : : 58

6.3 Change in two-jet inclusive and three-jet inclusive HT from re-vertexing. 59

6.4 Change in �3+�2+

from re-vertexing. : : : : : : : : : : : : : : : : : : : : : : 59

6.5 Single interaction fraction vs instantaneous luminosity and Instantaneous

luminosity distribution. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 60

6.6 �3+�2+

vs Instantaneous luminosity : : : : : : : : : : : : : : : : : : : : : : : 62

6.7 Underlying event : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63

6.8 Jet energy scale response function. : : : : : : : : : : : : : : : : : : : : : : 64

6.9 Jet energy scale response function (crossed) : : : : : : : : : : : : : : : : : 65

6.10 �3+�2+

vs HT for di�erent energy scale choices. : : : : : : : : : : : : : : : : 66

xii

6.11 Systematic errors. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 68

6.12 Cross section ratio showing statistical and systematic errors. : : : : : : : 69

7.1 JETRAD calculation of �3+�2+

vs HT . : : : : : : : : : : : : : : : : : : : : : 73

7.2 A. Splitting and merging of jets in the jet reconstruction algorithm. B.

Use of Rsep in Monte Carlo jets. : : : : : : : : : : : : : : : : : : : : : : : 75

7.3 Di�erence in the number of reconstructed jets vs D for a cone of radius 0.7 76

7.4 The fractional di�erence betweenRsep =1.3 and 1.2 for the three-jetHT distribution

(A) and the ratio �3+�2+

(B). : : : : : : : : : : : : : : : : : : : : : : : : : : 76

7.5 The percentage change in HT and�3+�2+

as we increase or decrease Rsep . : 77

7.6 �3+�2+

from JETRAD with di�erent parton distribution functions : : : : : : 78

7.7 Run 1a jet ET fractional resolution. : : : : : : : : : : : : : : : : : : : : : 80

7.8 Jet pseudorapidity (�) distribution for JETRAD Monte Carlo jets. : : : 81

7.9 Fractional percentage di�erence between smeared and raw jet pseudora-

pidity distributions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 81

7.10 Average jet ET vs jet pseudorapidity (�). : : : : : : : : : : : : : : : : : : 82

7.11 Comparison of the ratio �3+�2+

calculated with JETRAD, with and without

the smearing of jet ET . : : : : : : : : : : : : : : : : : : : : : : : : : : : : 82

7.12 The fractional di�erence between the nominal smear and the Hi and Lo

smearing. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 83

7.13 Jet-�nding e�ciency in di�erent pseudorapidity regions. : : : : : : : : : 85

7.14 Percentage error in �3+�2+

as a function of HT due to jets not reconstructed. 85

7.15 Errors in the theoretical prediction. : : : : : : : : : : : : : : : : : : : : : 86

7.16 JETRAD calculation of �3+�2+

using two di�erent renormalization scales for

soft jet emission. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 88

7.17 The average value of the third jet ET as a function of the HT of an event. 88

7.18 A �2 comparison of data with two theory prescriptions. : : : : : : : : : : 89

7.19 �2 di�erence between data and theory for di�erent hard scales. : : : : : : 90

7.20 The cross section ratio for three di�erent soft-scale choices. : : : : : : : : 91

xiii

7.21 �2 di�erence between data and theory as a function of the coe�cient of

the hard scale. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 92

7.22 �2 di�erence between data and theory as a function of the coe�cient of

the hard scale (�), for jet ET thresholds of 20 (a), 25 (b), 30 (c), and 40

GeV (d). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 93

7.23 �2 di�erence between data and theory, vs the hard scale, � < 2:0 : : : : : 95

B.1 Sample data and theory values to describe the covariance matrix �2 cal-

culation. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 106

xiv

Foreword

The D� collaboration consists of more than 400 physicists and students from 48

institutions from around the world. The University of Rochester has been active in

the following areas of D�: calorimeter design and calibration, data acquisition and

detector-monitoring shifts during collider runs, liquid argon purity monitoring, o�ine

data reconstruction, data analysis in the Top, QCD, W=Z, and New Phenomena physics

groups, and scintillating-�ber-detector design and development for the detector upgrade,

including characterization measurements of visible light photon counters.

I participated in the detector monitoring shifts, the calibration of the Level 1 calorime-

ter trigger, and the o�ine data reconstruction e�ort. For the latter I wrote and adapted

scripts to process the data �les and monitored the data ow through the reconstruction

system. I also acted as an administrative liaison between the QCD physics group (end-

users) and the O�ine Computing Production Board (OCPB). My analysis project has

been the study of cross-section ratios of multijet events.

1

Chapter 1

Introduction

When protons and antiprotons are made to collide head-on at high energies one sees

sprays of hadrons (pions, protons, neutrons, etc.) emerging from the interaction point

and clustering together in groups. These clusters are called jets. These jets frequently

emerge perpendicular to the path of the colliding proton and antiproton, and the expla-

nation of this behavior comes in the form of the quark model, where the proton (and

antiproton) is postulated to be composed of point-like charged objects called quarks.

Quantum Chromodynamics (QCD) is the theory of the interaction of quarks and gluons,

the latter being the �eld particle exchanged in an interaction between quarks. The above

interaction can then be described, for example, in terms of the annihilation of a quark

in the proton and an antiquark in the antiproton into a gluon, which then \decays" to a

quark-antiquark pair. The two outgoing quarks travel a short distance (. 1 fm) before

undergoing a hadronization process, whereby the quarks combine with other quarks to

form hadrons.

Sometimes, an outgoing (or incoming) quark (Figure 1.1) radiates a gluon at a large

angle (with respect to the quark's direction) and this gluon also hadronizes to become a

separate jet. This would yield a three-jet event.

It is possible to calculate the probability or cross section to produce two-jet �nal

2

pp

jet 2

jet 1 jet 1

p

A two-jet event A three-jet event

jet 3jet 2

p

Figure 1.1: Two-jet and three-jet events.

states and three-jet �nal states using QCD and perturbation theory. The goal of our

study is to measure the ratio of the inclusive-three-jet cross section to the inclusive-two-

jet cross section

�3+�2+

=�( pp ! n jets +X ; n � 3)

�( pp ! m jets +X ; m � 2)

as a function of the total transverse energy� of all jets in an event. (X denotes other

products of collision which we ignore.) We use the symbol HT to denote the sum of the

jet ET s:

HT =Xjets

ET

We compare the value of �3+�2+

as a function of HT for data and predictions from pertur-

bative calculations up to order �3s (Next-to-leading order).

One application of this study is in the design of a trigger for weak boson scattering

at the Large Hadron Collider to search for the Higgs particle. The second motivation

of this analysis is to determine a preferred renormalization scale for the measurement of

the ratio of cross sections.

�Transverse energy, ET , is de�ned as E sin � where � is the angle between the jet's axis and

the beamline.

3

1.1 Organization of thesis

I �rst outline the theory of Quantum Chromodynamics in Chapter 2. In Chapter 3 I

describe the Tevatron accelerator and the D� detector, and in Chapter 4 I describe data

acquisition and event reconstruction. Chapters 5 and 6 comprise the analysis of the data,

including the uncertainties in the measurement. In Chapter 7 I introduce the theoretical

predictions from Monte Carlo calculations and describe the quantitative comparisons

with data. In the last chapter I summarize the comparisons between data and theory.

4

Chapter 2

QCD

2.1 Introduction

In this chapter, I outline the nature of the strong interactions as formulated in the

Standard Model. I then discuss the application of Quantum Chromodynamics to hadron-

hadron collisions and jet production. Lastly, I describe the test of the renormalization

scale to be performed in later chapters.

2.2 Beginnings of QCD

The theory of strong interactions started with the need to explain why protons in a

nucleus did not repel each other and thereby make the nucleus fall apart. A force called

the strong force was postulated to keep the protons together despite their electrical

repulsion. The �rst �eld theory of strong interactions was due to Yukawa in 1934, in

which the attractive force between protons at short distances was attributed to the

exchange of a pion, discovered subsequently in 1947 [1].

Over the following two decades experiments at accelerators found many new particles.

The classi�cation of these particles came in 1961 with the EightfoldWay (Gell-Mann and

Ne'eman). Further simpli�cation came with the quark model of Gell-Mann and Zweig,

5

in 1964. The quark model holds that all baryons (e.g., proton) and mesons (e.g., pion)

are made up of quarks [2].

Other high-energy experiments were performed (1970s) with electron beams on pro-

ton targets, so-called deep inelastic scattering (DIS) experiments. These indicated that

the proton had a sub-structure when probed with an electron, the way atoms were long

ago found to have a nuclear sub-structure when probed with alpha particles. The name

given to the constituents of protons was partons. The quarks of hadron spectroscopy were

candidates for the partons found in DIS experiments. It appears that these constituents

cannot exist as free particles, but are con�ned within hadrons. This phenomena is called

con�nement. Furthermore, high energy lepton-hadron scattering experiments showed

that the interaction with partons is well-described by a model where the act as non-

interacting point-like objects within hadrons (the parton model.) More data revealed

that much of the nucleon mass had to be attributed to the presence of neutral partons

(later called gluons). The drop in the strength of the interaction among partons with

decreasing distance is called asymptotic freedom, and arises as a natural consequence of

QCD. QCD in its current form was proposed around 1974 [3].

2.3 The Standard Model

The Standard Model is the currently accepted theory of the interactions between matter

at the elementary particle level. It encompasses electricity, magnetism, strong, and weak

interactions. It does not include gravity as yet. The Standard Model is based on the

gauge group SU(3)color � SU(2)�U(1)Y , representing, respectively, strong interactionsor QCD, Weak Interactions, and Quantum Electromagnetism (QED).

The elementary particles that interact according to the Standard Model are quarks

and leptons. The mediators of the forces are the gauge bosons: photons ( ), W , Z, and

gluons (g). Composite particles are made up of either a quark-antiquark pair (mesons)

or a bound state of three quarks (or three antiquarks) called baryons (Table 2.1)

6

Table 2.1: Classi�cation of matter. The composite particles are at the bottom of thetable.

Matter

Fermions Bosons

Leptons Quarks Mediators

�e �� �� u c t W Z g

e � � d s bHadrons

Baryons Mesonsn p � �

The �rst sector of the Standard Model that was fully developed is electromagnetism,

or Quantum Electrodynamics. It describes the interaction of electrically charged par-

ticles via the exchange of photons. QED is based on the quantization of Maxwell's

Equations, and describes processes such as the Coulomb interaction, pair-production,

pair annihilation, and Compton scattering (e� + ! e� + ).

All quarks and leptons participate in weak interactions, mediated by the charged

W boson or the neutral Z boson. An example of a charged interaction is muon decay

(�� ! e� + �� + �e). Weak interactions also govern the process of neutron beta decay

(n! p+e�+�e). The electromagnetic and weak interactions are now uni�ed, and form

the electroweak theory, in which the electromagnetic and weak interactions are simply

di�erent manifestations of a single electroweak interaction.

The gauge boson of the electromagnetic interaction, the photon, is massless, while

the weak interaction bosons, the W+;W�; and the Z boson, are massive. (The QCD

gauge boson, the gluon, is also massless.) One explanation for this unsymmetrical state

is provided by the Higgs mechanism [4]. The Higgs mechanism is responsible for the

masses of the W� and the Z0, as well as the masses of the quarks and leptons. A Higgs

particle (or two) is required for most formulations of the model, and has been the object

of searches in modern particle colliders [4].

7

2.4 QCD

QCD describes the strong interactions between quarks as mediated by gluons. It is

a gauge �eld theory based on the group SU(3). SU(3) or SU(3)color is the group of

unitary transformations on color quark �elds. Requiring local invariance under color

transformations results in a law of conservation of \color charge," and also the presence

of a �eld (gluon �eld in this case) and its associated quantum particles (gluons). Quarks

carry color as well as electric charge. The three kinds of color \charge" are usually termed

Red, Green, and Blue. Antiquarks carry \anticolor." Color is exchanged between quarks

via eight gluons. The six known quarks are listed in Table 2.2 [5].

Table 2.2: List of Quarks.

Name Symbol Charge [e] Mass

Down d �13 2-8 MeV/c2

Up u +23 5-15 MeV/c2

Strange s �13 100-300 MeV/c2

Charm c +23 1.0-1.6 GeV/c2

Bottom b �13 4.1-4.5 GeV/c2

Top t +23 172.1�5.2 GeV/c2

Mesons are formed from a combination of two \valence" quarks, (qq) while baryons

are formed from three valence quarks (q1q2q3). In each case, the colors combine to form

a \colorless" object. Another way of saying this is that all free and isolated particles

are color singlets. The binding of nuclei in atoms is ascribed to the nucleon-nucleon

attraction that is due to the long-range interaction of bound quarks in one nucleon with

bound quarks in the other.

The strength of the color force depends on the distance scale of the scattering process.

At short distances and short time scales (high-energies), the QCD force is weak, which

provides the basis for applying perturbation theory. However, at longer time scales and

large distances (low energies), the strong force is indeed strong, where it accounts for

the binding of quarks within hadrons [6]. An indication of this behavior comes from the

8

form of the strong coupling (�S ) in the leading log approximation [3]:

�S(Q2) =

12�

(33� 2nf ) ln(Q2=�QCD2)

(lowest order)

with nf the number of quark avors (types) appropriate for the process. The parameter

�QCD is not predicted by QCD, but can be obtained from experiment. We can think of

�QCD as the natural scale of QCD. Q2 is the square of the momentum transfer in the

interaction. When Q2 is much larger than �QCD , the e�ective coupling of quarks to

gluons is small, and the approximation in which quarks are non-interacting is appropri-

ate. For Q2 of order �QCD , quarks and gluons form strongly-bound states of hadrons,

and this corresponds to the non-perturbative regime of QCD [7].

2.5 Perturbative QCD

Factorization theorem

A basis of perturbative QCD is the factorization theorem. Factorization refers to the

separation of the long-distance (low momentum transfer) and short-distance (high mo-

mentum transfer) parts of the interaction. The factorization theorem can be expressed

via the schematic of a quark-quark interaction in pp collisions in Figure 2.1 and the

equation

� =X

partons ij

Zdx1dx2 fh!i(x1; �

2F )fh!j(x2; �

2F )�ij

��S; Q

2=�2F�

� is the partonic (qq,qg,gg) point cross section. The fh!i are parton distribution func-

tions (PDF) that characterize the momentum fractions xi of the hadron h carried by the

parton i. The �F is the factorization scale, which can be thought of as the point where

one de�nes a separation between the short-distance and long-distance regimes. Parton

distribution functions are de�ned for 0 < x < 1, and give the probability of �nding some

9

x1P1

a

x2P2

b

fi(x1)

fj(x2)

σ∧

ij (αs)

Dh←k(z)

c

h1

h2

h3

d

jet

Figure 2.1: QCD description of a hard scattering process between hadrons h1 and h2with four-momenta P1 and P2. The partons from h1 and h2 have momentum fractionx1 and x2 respectively. Partons a and b scatter to partons c and d. Parton c hadronizesto form hadron h3 while parton d produces a jet.

parton a inside hadron h with a momentum fraction x. The parametrizations of f for

di�erent partons a and hadrons h are taken from collider and �xed-target experimental

data [8]. As shown in Figure 2.1, partons of momentum fraction x1 and x2 undergo a

point scatter into �nal state partons.

There are currently three main groups who perform \global �ts" to experimental

data to extract PDF's. These are:

� Martin, Roberts, and Stirling (MRS) [9]

� Gluck, Reya, and Vogt (GRV) [10]

� The CTEQ collaboration [11]

10

Parton distribution functions that are measured at some given interaction scale Q0 can

be used to determine the values at another scale Q via the Dokshitzer-Gribov-Lipatov-

Altarelli-Parisi (DGLAP) evolution equations [12]. These are sets of coupled integral-

di�erential equations that describe the Q2-dependence of quark and gluon distribution

functions [13].

d

d logQ2fh!i(x;Q

2) =�S

2�

Z 1

x

dy

yfh!i(y;Q

2)Pqq

�x

y

�

This equation is for quarks only. Similar equations exist for g ! q, q ! g, and g ! g.

The splitting function Pqq(xy ) is the probability of a quark to emit a gluon, and thereby

becoming a quark with momentum reduced to x=y of its original value [14].

Hard scattering matrix elements

The partonic cross section � is calculated using the Feynman rules for QCD. For two-

to-two parton scattering in hadron-hadron collisions, the relevant diagrams are shown

in Figure 2.2 [15]. The square of the invariant matrix element for two-body scattering

of massless partons are listed in Table 2.3. Here s = (p1 + p2)2, t = (p1 � p3)

2, and

u = (p2� p3)2 are the Mandelstam variables for the hard scattering subprocess [15]. (p1

and p2 are the momenta of the initial-state partons, while p3 and p4 are the �nal-state

momenta.)

The matrix elements for quark-quark scattering have been calculated to O(�3S). Some

of the diagrams that contribute at O(�3S) are shown in Figure 2.3. For the two-to-three

parton scattering processes, we can divide these into four groups:

11

Figure 2.2: Leading Order (�2S) diagrams for two-body scattering. Lines represent

quarks and curly symbols represent gluons.

(A) q(p1) + q0(p2) ! q(p3) + q0(p4) + g(k)

(B) q(p1) + q(p2) ! q(p3) + q(p4) + g(k)

(C) q(p1) + q(p2) ! g(p3) + g(p4) + g(p5)

(D) g(p1) + g(p2) ! g(p3) + g(p4) + g(p5)

In process A, the quarks q and q0 are of di�erent avor, while the quarks in process B

12

Table 2.3: Invariant matrix elements for 2-to-2 scattering of massless partons.

ProcessPjMj2=g4

qq0! qq0 49s2+u2

t2

qq0 ! qq0 49s2+u2

t2

qq! qq 49( s

2+u2

t2+ s2+t2

u2)� 8

27s2

ut

qq! q0q0 49t2+u2

s2

qq! qq 49( s

2+u2

t2+ t2+u2

s2)� 8

27u2

st

qq! gg 3227

t2+u2

tu� 8

3t2+u2

s2

gg ! qq 16t2+u2

tu� 3

8t2+u2

s2

gq ! gq �49s2+u2

su� u2+s2

t2

gg ! gg 92(3 � tu

s2� su

t2� st

u2)

are identical in avor. The symbol g(k) represents a gluon with momentum k. All other

matrix elements for 2-to-3 parton scattering can be obtained by crossing ( a+ b! c+ d

is the same as a+ c! b+ d). For process A above, the square of the matrix element is

X���M(A)���2 = 4g6

9

s2 + s02 + u2 + u02

2tt0

!�8

3([14] + [23]) +

1

3[12; 34]

�

Here g =p4��S and

s = (p1 + p2)2; t = (p1 � p3)

2; u = (p1 � p4)2

s0 = (p3 + p4)2; t0 = (p2 � p4)2; u0 = (p2 � p3)

2

and the eikonal factor [ij] is

[ij] =pi � pj

pi � k k � pj

and �nally

[12; 34] = 2[12] + 2[34]� [13]� [14]� [23]� [24]

13

Figure 2.3: Diagrams for Next-to-leading order (�3S) two-body scattering. Lines repre-

sent quarks and curly symbols represent gluons.

The matrix elements in terms of s; t; u, etc., for processes B;C; and D are even more

complicated and are given in Appendix A [16].

Parton Showering and Jet production

The development of a �nal-state parton to a jet is divided into two steps: First the

parton shower, and then hadronization. The dividing line between the two steps is the

value of the momentum-transfer scale, usually taken to be of order 1 GeV. In the parton

shower, a quark or gluon can emit a gluon and decrease its momentum, or a gluon can

14

split into a quark-antiquark pair. This continues until the partons have energies of 1

GeV or so. At this point hadronization occurs, which converts the partons into observed

hadrons. The nearly collinear group of produced hadrons forms a macroscopic structure

called a jet. The hypothesis of local parton-hadron duality equates the �nal energy and

direction of a jet with that of the parton which gave rise to it [17].

The cross section for producing speci�c �nal-state hadrons can be studied to extract

the fragmentation functionDh k, which is analogous to the parton distribution functions

but gives the probability to produce a hadron h from parton k (see the upper-right leg

in Figure 2.1).

Jet variables

The colliding proton and antiproton beams in our experiment have the same energy.

However, the partons from each hadron do not carry the same momentum fraction,

resulting in a partonic center of mass that is boosted relative to the hadronic center of

mass. (The partons can also carry a small amount of \inherent" transverse momentum,

but this is usually attributed to soft-gluon radiation, and often neglected.) It is therefore

useful to work with quantities that transform simply with respect to boosts along the

beam direction. The traditional variables used are rapidity (y), transverse momentum

(pT ), and azimuthal angle (�) of the jet. For a jet with energy E and momentum p, the

rapidity (y) is de�ned as

y =1

2ln

�E + pzE � pz

�

the transverse momentum as

pT =qp2x + p2y

The transverse energy ET is frequently used in place of pT in cases where the mass of

the object can be neglected or when the mass of the particle cannot be measured. In

15

addition, the pseudorapidity � is usually de�ned in place of rapidity, again when the

mass of the object is much smaller than its transverse momentum. The pseudorapidity

(Figure 2.4) in terms of the polar angle � (with respect to the beam direction) is

θ=40

y η = 0

η = −1θ=90

o

o

o

θ=5.7

(beamline)z

η = 3

η = 1

Figure 2.4: Pseudorapidity.

� = � ln tan(�=2)

and transverse energy can also be written as

ET = E sin �

Although the de�nition of a jet is somewhat arbitrary, the same de�nition should be

used in experiment and in Monte Carlo in order to facilitate comparison with theoretical

predictions. The jet algorithm used in D� is known as the cone algorithm. Here, a jet

is a concentration of transverse energy inside a \cone" of radius R, where

R =p(��)2+ (��)2

By using � instead of � we ensure that the de�nition of a jet is invariant under longitudinal

boosts [15].

16

q

HW,Z

W,Z

W,Z

W,Z

q

qq

q

Figure 2.5: Higgs production via weak boson scattering.

2.6 Ratio of 3-jet and 2-jet cross sections

The goal of this measurement is to study the production rate of soft jets and to determine

the scale for such soft-jet emissions. An application of this study is in weak boson

scattering at LHC energies. In the process qq0 ! qq0H ! qq0WW (Figure 2.5) the event

signature consists of jets in the forward and backward (large j�j) region and nothing in thecentral (� � 0) region. The major backgrounds to this process are qq ! tt! W+bW�b

and qq ! W+W�. These backgrounds have a signature of jets in the central region.

The ability to veto central jets in the trigger can enhance signal-to-background for the

qq0 ! qq0WW process. Thus, an understanding of the production rate of soft jets in the

central region can aid in designing a trigger to capture these events. The cross section

ratio �3+�2+

is a measure of the probability for the emission of a soft jet in dijet events [18]

[19].

A measurement of �3+�2+

gives us an estimate of the production rate for soft jets, while

a comparison of �3+�2+

using either a hard scale (�R =PET ) or a soft scale (�R = E

(3)T )

in the Next-to-Leading Order calculation tells us whether the emission of soft jets is

governed by the hard scattering scale or the soft scale [20].

17

Chapter 3

The Tevatron and the D�

Detector

3.1 Introduction

Our study of pp interactions was done at Fermilab (Batavia, Illinois). The creation,

acceleration, and collision of protons and antiprotons was performed with the Tevatron

accelerator at Fermilab. The detection of the products of collision was done using the

D� detector. This chapter will describe the Tevatron and the D� detector.

3.2 The Tevatron

The Fermilab Tevatron is the largest of a series of accelerators working in sequence to

produce and collide protons and antiprotons at a center-of-mass energy of 1.8 TeV. The

main parts of the collider, in order of increasing beam energy, are: the Pre-Accelerator,

the Linac, the Booster, the Main Ring, and the Tevatron. They are listed in Table 3.1.

The layout of the accelerator is shown in Figure 3.1.

The �rst step in creating proton beams is the production of of H� ions in the Pre-

accelerator; it consists of a hydrogen ion source and an electrostatic accelerating column

18

Tevatron

Main Ring

DO detector

CDF

AO

BO

CO

DO

EO

FO

MR P Injection

Booster

PreAcc

LinacPBarDebuncher

PBarAccum

PBarTarget

Tevatron RF

Main Ring RF

PBar Injection

TevatronInjection

P and PBarAborts

PBar

P

Tevatron Extractionfor Fixed Target Experiments

Figure 3.1: The Fermilab collider (not to scale)

powered by a Cockcroft-Walton generator. The H� ions produced from a surface-plasma

magnetron are accelerated through a 750 keV potential and then fed to the Linac.

The Linac consists of �ve steel \tanks" arranged end-to-end with a gap between the

tube ends. An alternating electric �eld is applied inside the tanks. A charged particle is

accelerated when it is in the gap between the drift tubes. Inside the tubes it is shielded

from the �eld, and will simply drift. The frequency of the alternating electric �eld is such

that an accelerating �eld is present when the particle is in the gap and a decelerating

�eld is present when the particle is in the drift tube. As a particle travels down the

19

Table 3.1: Tevatron and associated accelerators

Name Type Start Energy End Energy DimensionPre-Accelerator Linear 0 keV 750 keV Height = 11 mLinac Linear 750 keV 400 MeV Length = 146 mBooster Circular 400 MeV 8 GeV Dia = 151 mMain Ring Circular 8 GeV 150 GeV Dia = 2000 mTevatron Circular 150 GeV 900 GeV Dia = 2000 m

linac its energy increases and so the drift tubes become increasingly longer to maintain

the same phase. In addition, the drift tubes contain alternate focussing and defocussing

quadrupole magnets to contain the lateral spread of the ions due to space charge and

RF e�ects. The ions have an energy of 400 MeV after leaving the Linac.

At the end of the Linac the H� ions pass through a carbon foil to strip o� the

electrons, leaving only protons. The protons are fed to a synchrotron called the Booster.

A synchrotron is a closed-orbit accelerator with magnets that bend the beam into roughly

circular orbits, using cavity resonators to increase the energy of the beam. As the beam

energy is increased the magnetic �eld strengths are correspondingly increased to keep

the particles in the same path. The Booster then raises the energy of the protons from

400 MeV to 8 GeV. For extraction \kicker" magnets are used to transfer the entire beam

in one turn.

The Main Ring is a 1000 m (radius) synchrotron that is capable of producing 400

GeV proton beams. As a feeder for the Tevatron is it used to raise the beam energy

to 150 GeV. The Main Ring and Tevatron RF systems are phase-locked using common

timing signals. The Main Ring RF system is then aligned with the Tevatron RF system

and the proton bunch in injected into a Tevatron bucket.

The production of antiprotons begins with the Main Ring. The protons are acceler-

ated to 120 GeV, extracted, and �red at a nickel target. Antiprotons with energies of 8

GeV are collected and stored in the Debuncher. Debunching is the process of reducing

20

the momentum spread of the antiproton beam. The antiprotons are then injected into

the Accumulator.

In collider mode, the antiprotons are extracted from the Accumulator and fed to

the main ring. Proton bunches are also injected into the main ring, and travel in the

opposite direction as the antiprotons. When the proton and antiproton bunches reach

150 GeV they are injected into the Tevatron where they continue to rotate in opposite

directions. The Tevatron is similar in many ways to the main ring, sharing the same

tunnel. The di�erence is in the use of superconducting magnets in the Tevatron. The

proton and antiproton bunches are accelerated to 900 GeV and are made to collide at

two points, where the CDF and D� detectors are located. Additional information on

the operation of the Tevatron can be found in Reference [21].

3.3 The D� detector

The D� (pronounced \D-zero") detector consists of three main parts that approximate

layers surrounding the interaction point. From the center outward, these are the Central

Detector, the Calorimeter, and the Muon Detector. A drawing of the detector showing

all three systems is in Figure 3.2.

3.3.1 The D� coordinate system

The coordinate system is de�ned such that the beams travel along the z axis, with

protons moving in the +z direction (south). The positive y direction is upward from the

center of the detector. Lastly the positive x is de�ned so the overall coordinate system

is right-handed (east). The polar angle � is de�ned from the +z axis. Another measure

of polar angle is the pseudorapidity �, de�ned as

� = �ln(tan(�=2))

21

D0 Detector

Muon Chambers

Calorimeters Tracking Chambers

Figure 3.2: The D� Detector

22

The azimuthal angle � is measured from the +x direction, toward the +y direction.

Particle and jet trajectories are usually given in �-� \coordinates."

3.3.2 Central detector

The Central Detector consists of the Vertex Drift Chamber (VTX), the Transition Ra-

diation Detector (TRD), the Central Drift Chamber (CDC), and two Forward Drift

Chambers (FDC). See Fig 3.3. There are a total of 4200 wires and 6080 channels in

the central detector. The innermost layer (Central Detector) is used for measurement

ΘΦ Central DriftChamber

Vertex DriftChamber

TransitionRadiationDetector

Forward DriftChamber

Figure 3.3: The Central Detector elements. Beam goes horizontally in the �gure throughthe center beamline.

of the interaction vertex (position in z, along the beamline), charged particle track

measurement, and ionization energy measurement to distinguish electrons from photon

conversion products ( ! e+e�).

The vertex drift chamber (Fig 3.4) consists of three concentric layers of cells with

wires running parallel to the beamline. Each cell has eight wires at di�erent radial

distances from the beamline and these determine the r-� position of a track. The z-

23

Figure 3.4: The layout of sense and grid wires in the Vertex Drift Chamber. The beamis oriented perpendicular to the page at the center of the arcs.

position is determined by reading the sense wires on both ends. The purpose of the

vertex drift chamber is the measurement of the interaction vertex.

The Transition Radiation detector is located just outside the vertex chamber and

is used to distinguish between electrons and hadrons. A charged particle traversing a

boundary between two dissimilar materials (the \radiator") will emit x-rays (the transi-

tion radiation). In the TRD, the radiator is in the form of nitrogen gas and the radiation

is detected using proportional wire chambers located downstream. The TRD has three

sets of radiator and PWC pairs. The x-ray detector is performed in two stages. In the

�rst stage the x-ray converts to an electron-positron pair ( ! e+e�). In the second

stage the conversion products and other particles are detected in a wire chamber.

The Central Drift Chamber (Figure 3.5) is part of the tracking system for the central

region (perpendicular to beamline). The CDC has four layers, each with 32 cells. Each

cell has seven sense wires read out at one end. Also there are two \delay lines" read

out at both ends. These delay lines propagate signals induced from nearest anode wire.

A measurement of di�erences in arrival times at two ends permit z-coordinate location.

24

Figure 3.5: A cross-sectional view of the Central Drift Chamber.

Forward drift chambers (Figure 3.6) are used for tracking charged particle trajectories at

small polar angles (almost parallel to beamline). The FDC has three layers of drift cham-

bers, one Phi layer with sense wires oriented radially and two Theta layers surrounding

the Phi with sense wires approximately in a circular pattern around the beamline. The

parameters of the Central Detector elements are shown in Table 3.2.

Table 3.2: Central Detector Parameters

Central Detector

Subsystem Resolution Radius Height Purpose

Vertex Drift ChamberVTX

50 �m 4-16 cm 104 cm Determine interactionvertex

Central Drift ChamberCDC

150-200 �m 50-75 cm 184 cm Charged particletracking

Forward Drift Cham-ber FDC

150-200 �m 10-60 cm 40 cm Charged particletracking

Transition RadiationDetector TRD

�-e re-jection of50

16-50 cm 184 cm Identi�cation of elec-trons and pions

25

Figure 3.6: An expanded view of the Forward Drift Chambers, showing the three layersof drift chambers. The arrow shows the beam orientation and points to the center of thedetector.

3.3.3 Calorimeter

The calorimeter is used to measure the energy of electrons, photons, and hadron jets.

It is designed to cover nearly the full solid angle around the interaction region for good

measurement of missing ET and good coverage of the far-forward region (parallel to the

beamline). Figure 3.7 shows a cutout view of the calorimeter, showing the three cryostats

which contain the Central Calorimeter (CC) and the two End Calorimeter (EC).

As a particle travels through the calorimeter, it interacts with the calorimeter ma-

terial and deposits its energy into the calorimeter material. A small fraction of this

deposited energy is detectable as a signal that is proportional to the incident particle's

energy, allowing its measurement.

The D� calorimeter is made up of uranium and liquid argon layers (see Figure 3.8).

The uranium acts as an absorber (inactive layer) which causes the primary particle to

interact with it (Bremsstrahlung) and produce daughter particles which then ionize the

liquid argon (active layer). A voltage of 2 kV is maintained across the liquid argon,

between the uranium plate and an electrode. The negative ions produced drift toward

26

1m

D0 LIQUID ARGON CALORIMETER

CENTRAL CALORIMETER

END CALORIMETER

Outer Hadronic(Coarse)

Middle Hadronic(Fine & Coarse)

Inner Hadronic(Fine & Coarse)

Electromagnetic

Coarse Hadronic

Fine Hadronic

Electromagnetic

Figure 3.7: The D� calorimeter showing di�erent segmentation.

G10 InsulatorLiquid Argon

GapAbsorber Plate Pad Resistive Coat

Unit Cell

Figure 3.8: A schematic of a calorimeter cell.

27

the electrode, causing more ionization of the liquid argon. This produces a small current

which is ampli�ed and recorded. The collected charge is proportional to the energy of

the incident particle.

To measure the position of the showers produced by charged particles, the combina-

tion of uranium and liquid argon is divided into segments or \cells", each instrumented

independently (Fig. 3.9). The segmentation is done so that the cells are roughly the

Figure 3.9: A view of a quarter of the calorimeter showing the �ne longitudinal andtransverse segmentation. The beams travel horizontally through the beam pipe shownat the bottom of the �gure and the interaction region is approximately in the lower leftcorner of the �gure.

same size in � and � (but not in x and y). In addition, there is segmentation in the

radial direction to determine the \depth" of the shower. This is useful in distinguishing

between electromagnetic objects (photons, electrons) and hadronic objects (pion, eta

meson, rho meson, etc.) The choice of segmentation results in a natural grouping of

cells of the same � and � (i.e., same \direction") but of di�erent layers. This group

28

Table 3.3: Calorimeter Parameters.

CalorimeterSpatial resolution 0.8-1.2 mm

Radius 75-500 cmHeight 306 cm (CC) 263 cm (EC)

Energy resolution 15%=pE(EM); 50%=

pE(Had); E in GeV

No. of channels 50,000

of cells with the same direction from the center of the detector forms what is called a

tower. This concept of a tower will be used later in triggering and in the jet-�nding

algorithm. Cells have a size of 0.1 � 0.1 in �-� (about 3 in. � 3 in.) in the central

region (perpendicular to beamline at the interaction region) and 1.5 in. � 0.1 in. in the

forward region (parallel to beamline).

The calorimeter surrounds the Central Detector on all sides. To allow access to the

Central Detector the calorimeter was split into three parts, the central calorimeter (CC)

and two end calorimeters (EC). In order to provide good spatial resolution while still

keeping the volume (and hence cost) small, both the CC and EC have di�erent modules

with increasing distance from the interaction region: an electromagnetic (EM), a �ne

hadronic (FH) and a course hadronic (CH) section. The EM section uses thin uranium

plates. The �ne hadronic uses thicker uranium plates and the CH uses copper or stainless

steel. The hadronic sections are further away from the interaction region since hadrons

typically produce showers later than electromagnetic objects do.

Also, since electromagnetic objects (photons, electrons) produce smaller showers the

segmentation in the EM calorimeter is �ner than in the hadronic calorimeter. The

position resolution of the calorimeter for isolated electrons is between 0.8 and 1.2 mm

and varies as E�1=2 where E is the particle energy. The calorimeter parameters are

summarized in Table 3.3.3.

In order to keep the liquid argon at low temperatures, the calorimeter modules are

29

placed in a double-walled cryostat. The cryostats are sealed, but have ports to exchange

liquid helium and to provide a path for the ampli�er signals to exit. Additional ports

exist to pass high voltage, temperature, and purity monitoring signals. The calibration

of liquid-argon response to energy deposition is dependent on the purity of the liquid

argon, so a purity monitor is employed.

3.3.4 Muon system

The outermost detector system in D� is the muon system. It is located outside the

calorimeter. It is designed to measure muon momenta and charge. It consists of a

toroidal magnet to de ect the muons and proportional drift tube chambers located before

and after the magnet to record track coordinates. The track position is measured once

before entering the magnetic �eld and twice afterwards. This is combined with vertex

and tracking information from the Central Detector to determine the trajectory.

For a full description of the D� detector see Reference [22].

30

Chapter 4

Data Acquisition, Trigger, and

O�ine Event Reconstruction

4.1 Introduction

The signals that come out of the detector are stored for later analysis. Before they are

written to magnetic tape a �ltering system is used to remove uninteresting events and

reduce the signal rate to a manageable level. This is the purpose of the trigger and data

acquisition (DAQ) system. The DAQ system (Figure 4.1) is a �lter system in three stages

10-20

50 kHz300 kHz 1-2 Hz

Level

1Tape

Processing Time

0Detector

Level

Level

2

From the

10 kHzRate

200 Hz

100 Hz

Level

1.5

µs 100 - 200 ms900 ns 132 ns

Figure 4.1: D� trigger

that takes a quick look at each event to decide whether or not to save the information

on that event. It has three levels of event characterization, called the Level 0, Level 1

31

and Level 2 triggers. In the next section I will describe in detail the Level 1, 2, and 3

triggers, followed by the o�ine data reconstruction, and ending with a description of the

jet-�nding algorithm used in the data reconstruction.

4.1.1 Level 0

The Level 0 trigger (Figure 4.2) uses the Level 0 detector, which is a pair of scintillator

hodoscopes surrounding the beampipe and located at the inside face of the end calorime-

ter. The active elements of the hodoscopes extend radially to 45 cm from the beampipe

EC CC EC Beamline

Calorimeter Modules

Figure 4.2: Level 0 Detectors. Shaded areas show the scintillator hodoscopes.

and give coverage in the region 1:9 < � < 4:3.

In addition to its use in the Level 0 trigger, it is also used tomeasure the instantaneous

luminosity (the number of particles in a beam passing a boundary, per unit area, per

unit time) that is \seen" by the whole trigger system. This is done by measuring the

rate of interactions and using the known cross section for inelastic pp collisions. The

Level 0 detectors are also used to measure the location of the interaction point along the

beamline for use in the Level 1 and Level 2 triggers in calculating transverse energy.

A coincidence between the two Level 0 detectors indicates an inelastic collision and

this prompts the next trigger stage to check the event. The e�ciency of the Level 0

detector is 99% for non-di�ractive inelastic collisions [23].

32

ELECTRON

FRONT-END

EACH DATA CABLETO BE READ ONSELECT THE CRATES

GLOBAL ENERGY AND

CANDIDATES

DETECTOR

DETECTOR

TABLE OF

ABOVE Pt THRESHOLD

Z POSITION,BEAM GAS,

TWO INTERACTIONS

MOMENTUM SUMS,

SUPERVISOR

EVENT READOUT

CRATE DATA

SEQUENCERS

DATA

FRONT-END BUSY SIGNALS

HOLD TRANSFER SIGNALS

EVENT THAT PASS LEVEL 2 TRANSFERED TO THE HOST

CLUSTER CUTSLOCAL ENERGY

TO RECEIVE THE EVENTLEVEL 2 NODE

ENABLES A

THE SEQUENCERS

LEVEL ONE TRIGGER DATA BLOCK

ACQUISITION

PROCESSOR

CRATES

FRONT-END

FARM

LEVEL TWO

ON DATA CABLES

DATA CABLE

HOSTCOMPUTER

TRDDETECTOR

DATA CABLE BUFFER DRIVERS

LEVEL TWO SUPERVISOR

TRIGGER

WORK

LEVEL ONE

TRIGGERCONTROL

CALORIMETERCALORIMETER

MUON

LEVEL ONE

TRIGGER1280 Hadronic

TRIGGER TOWERS

FRAME-

TRIGGERMUON

START DIGITIZATION SIGNALSMUON FOUND

LEVEL ONE

TRIGGER

TRIGGERLEVEL ZERO

TRD

LEVEL ZERO

CONFIRMATION OF

16 BIT TRIGGER NUMBER

ETHERNET

AND MONITORING INFORMATION

PROGRAMMING, CONTROL

TRIGGER TOWERS

1280 EM

ELECTRON CANDIDATES

Figure 4.3: The Level 1 Trigger System.

4.1.2 Level 1

If an event passes the Level 0 trigger it is fed to the Level 1 trigger. A schematic of the

Level 1 system is in Figure 4.3. The Level 1 trigger uses information from the calorimeter

trigger and muon system trigger.

The signals from the di�erent detector elements in the calorimeter are sampled and

a quick calculation is made of quantities such as the total energy, missing ET , and

calorimeter trigger tower energies. The calculation of transverse energy uses an estimate

of the z-position of the vertex from the Level 0 detector. These quantities are used

either alone or in combination to form conditions that are required to be met, or else the

event is dropped from further consideration. There are 32 di�erent conditions that can

be satis�ed by the Level 1 trigger, with each condition being made up of one or more

sub-conditions from the calorimeter or the muon system [22].

The decision-making in the combined L0 and L1 trigger has to be done within the 3.5

�s time-between-bunch-crossing in order to incur no deadtime. There is also a Level 1.5

33

trigger which requires more than one time period between bunch crossing to complete

its calculation. The Level 1.5 trigger was not used in this analysis.

4.1.3 Level 2

When an event passes any of the 32 Level 1 conditions it is considered to have passed

the Level 1 trigger and is handed o� to the Level 2 system. The Level 2 system is a

software-based decision making system that applies more sophisticated tests to events.

Because of the longer time scale for calculations in Level 2, an event data bu�ering

and distribution system is used to allow several Level 2 calculations (one for each event)

to be performed concurrently. The distribution system sends an event to one of 50 VAX

workstations that perform the Level 2 calculations.

In Level 2, there are 128 di�erent criteria (called �lters) that can be satis�ed by

an event for that event to be kept and recorded. These criteria are made up of event

quantities similar to Level 1 quantities but are more complex. The Level 2 �lters also

require the passing of a speci�c Level 1 trigger as part of its requirements list. [22].

For this analysis, the Level 2 �lters used are the \single inclusive jet" triggers. These

require that the event has one or more jets above a minimum jet ET . Because the

ET distribution of jets is steeply falling, several inclusive-jet �lters with di�erent trigger

thresholds are used to sample the entire spectrum with good statistics.

Some Level 2 �lters pass events at a rate that is still too high for writing to tape.

In these instances a fraction of events are simply thrown away with the assumption that

this is done without bias. When calculating the rates for a trigger, the recorded number

of events is scaled accordingly. When an event passes the Level 2 �lter it is written to

8mm tape. These events constitute the \raw data."

34

4.2 O�ine processing

4.2.1 Farm

A small fraction of events are analyzed immediately in the Online system in order to

check that both the detector and the data acquisition systems are working properly.

However most of the data is written to tape and its reconstruction done in a separate

system called the o�ine system. The main reconstruction of the raw data is done a Unix

\farm."

The Unix farm was a group of networked Silicon Graphics or IBM workstations

running the event reconstruction program called RECO. The basic unit of the farm is an

FARMDZERO

RAW Data Tape

RECOblanktapes

File Server

DSTftp

D0FSSTA

Figure 4.4: O�ine Data Processing Farm

\I/O node" which acts as the event server and eight \worker nodes" which run RECO

and return the output back to the I/O node. Process control such as tape-to-disk (and

vice versa) spooling and �le naming are done on the I/O nodes. The input to the farm

are raw data tapes and the output is the same data in a di�erent, more manageable

format. These formats may go through one or more additional reprocessing steps in

order to group together events that are of interest to a given physics analysis subgroup

35

or to strip o� unneeded information to reduce the �nal �le size. A schematic of the farm

is shown in Fig 4.4.

The reconstruction program produces two output �les: STandard Output (STA, 600

Kbyte/event) and Data Summary Tape (DST, 20 Kbyte/event). The STA �les contain

all the information in the RAW data as well as parameters of reconstructed objects

(electrons, photon, jets, muons, etc.) The DST �les contain a stripped-down version of

the STA, with the most important information in an event. Frequently some low-level

information is stripped o� to produce compressed versions of the above called microSTA

and microDST. In this analysis the microDSTs are reduced even further to produce

ntuples (Section 4.2.3). [22].

4.2.2 Event reconstruction

The reconstruction program RECO converts the signals from the di�erent detector sys-

tems (Central Tracking, Calorimeter, and Muon System) into candidate objects such as

electrons, photons, muons, jets, and taus. It also calculates event quantities such as miss-

ing ET and total ET , interaction vertex location, Central Detector tracking information,

and muon tracking.

The z-position of the point where the partons from the proton and antiproton collide

is called the interaction vertex. Accurate measurement of this vertex is needed as other

measured quantities depend on it. The vertex is measured in several ways for di�erent

purposes. A quick determination of the vertex from the Level 0 detectors is used in

triggering by the Level 2 trigger. A more precise and accurate vertex measurement is

taken using the Central Drift Chamber (CDC). The measured vertex is used to calculate

particle transverse energies and directions. A requirement that that the vertex be in

the range �100 cm and +100 cm is made in the reconstruction process. Events whose

measured vertex is beyond this range are not reconstructed. This is to ensure that the

hard scatter occurs well within the detector.

36

Jet algorithm

A key element of RECO is the jet reconstruction algorithm. Interactions which produce

�nal-state jets will deposit energy into the calorimeter. Because the calorimeter is divided

into cells, a procedure for choosing which cells will be considered in de�ning the jet is

needed. This is the purpose of the jet algorithm.

In this analysis a �xed-cone algorithm is used. (This algorithm is a slight variant

of the so-called Snowmass jet algorithm [24].) Here a jet is de�ned by the total energy

deposited within a \cone" in �-� space. The size of the cone is given by its radius R:

R =p�2 + �2

A reconstruction cone radius of 0.7 is used in this analysis because it contains most of

the energy of a jet and is a standard size used by the D� and CDF experiments with

well-understood systematics [25]. It is also important for the algorithm to be infrared-

safe when used in theoretical calculations, to facilitate comparisons between a measured

quantity and a calculated theoretical prediction [6].

The cells in the calorimeter with the same � and � (but di�erent layers) are grouped

together into towers. The tower structure is shown in Figure 3.9. The jet �nding process

begins with listing all the calorimeter towers which have energy deposits in them. These

are sorted in decreasing ET . The largest ET tower is used as a seed. Any towers adjacent

to it that have an ET greater than 1 GeV are associated with that seed tower (Figure 4.5).

This continues with other adjacent towers up to a maximum 0.3 units in � or �. The

product of this step is called a precluster. The towers included with the �rst seed are

removed from the tower list and the remaining tower with the highest ET becomes the

next seed. The process of associating adjacent towers is repeated for this and later seeds.

This continues until no seed with ET above 1 GeV remains [23].

Using the list of preclusters from the previous step, the algorithm calculates an ET -

weighted axis for that precluster. A cone in �-� space is \drawn" around that axis and

37

Seed tower

η,φ

Rη,φ

Candidate tower

Figure 4.5: Preclustering.

all towers within that cone constitute a jet. A new ET - weighted axis is calculated from

all associated towers, and the process of drawing a cone is repeated until the jet axis

changes by less than 0.001 in �-� space or the number of iterations exceeds 50. If the

jet has an ET greater than 8 GeV it is stored and tested for splitting and merging (see

below). This process is repeated for all preclusters.

As each jet in the previous step is constructed independent of other jets, it is possible

for two jets found above to share one or more calorimeter towers. This is resolved by

the Split/Merge process. If a jet (starting with the second) shares any towers with a

previously found jet, the two jet axes are compared. If they di�er by less than 0.01 in

�-� space then they are considered the same jet. This may happen due to round-o�

errors. The second jet is then dropped from the list. If the two jets are not identical,

then a decision on whether and how to divide the jets is made using the quantity f

38

de�ned as

f =ET shared

ET min

where ET shared is the transverse energy shared by the two jets, and ET min is the smaller

of the two jets' ET . If f � 0:5 then the two jets are considered separate and the shared

cells are assigned to one jet or another depending on which cell in the shared tower is

closest to a jet axis. If f is greater than 0.5 then the two jets are combined into one. The

towers from both jets are used to calculate an ET - weighted direction, which becomes

the direction of the \merged" jet.

Missing ET measurement.

The D� calorimeter is designed to completely surround the interaction region except

for the beampipe. For this reason, an imbalance in the momentum measurements is

attributed to the presence of very weakly interacting particles (neutrinos and muons).

The \missing energy" may be attributed to these particles. It is more common to measure

the missing transverse energy (E/T ). It is de�ned as the vector ET that balances out the

sum of all measured vector ET s. The measurement of missing ET is used to remove

contaminated events (Section 5.3.1).

4.2.3 Data structure

The end product of o�ine data reconstruction is a series of events, each consisting of a

set of quantities that describe the event. The variables that make up the data structure

of an event include such quantities as:

� The (serial) run number.

� The (serial) event number for this run.

� The instantaneous luminosity for the event.

39

� The Level 2 triggers the event passed (can be more than one).

� The z-position (along beamline) of the interaction point, as calculated by both

Level 0 and the Central Detector.

� The number of charged-particle tracks seen by the tracking chamber

� The number of jet candidates found by the reconstruction algorithm

� The total energy deposited into the calorimeter.

� The measured missing ET of the event.

In addition, there are also variables that relate to speci�c objects within an event.

For example, for each jet found the following jet information is recorded:

� The energy of the jet

� The transverse energy of the jet

� The direction of the jet (�; �; �)

� The jet energy before rescaling (Section 6.6)

� The fraction of the jet energy deposited in di�erent calorimeter modules (Electro-

magnetic, Hadronic)

The �nal data set takes the form of \n-tuples." An ntuple is a list of identical data

structures, one for each event [26]. The exibility and compactness of the ntuple data

structure allows the selection of subsets of the data sample using selection criteria based

on one or more variables.

40

Chapter 5

The Data and Measurement of

the Cross Section Ratio

5.1 Introduction

In this chapter I describe in detail the measurement of the cross section ratio. I begin

with a description of the data sample, followed by the selection criteria used to clean up

the sample. Lastly, I outline the method used to combine the separate data sets (from

di�erent triggers) into one.

5.2 Data sample

The data used in the analysis is from the 1992-1993 Collider run. The total data sample,

consisting of 40542 events, was recorded using the \single-jet inclusive" triggers. These

triggers are designed to collect events that contain at least one hadron jet with transverse

energy (ET ) above a particular threshold value. Five di�erent thresholds are used: 20,

30, 50, 85, and 115 GeV. These are named JET 20, JET 30, JET 50, JET 85, JET 115.

A distribution of the event jet multiplicity (number of jets in the event) for events from

all �ve triggers is shown in Figure 5.1 and tabulated in Table 5.1.

41

Table 5.1: Jet multiplicity

No. of jets Events

2 16,3843 16,2014 5,9225 1,5846 3527 848 129 3

Figure 5.1: The distribution of jet multiplicities.

42

5.3 Event and jet selection criteria

The output of event reconstruction is a set of candidate events, with each event containing

one or more candidate jets. Event and jet selection criteria are applied to \clean up" the

data sample. The selection criteria fall into two categories: event selection criteria and jet

selection criteria. Event selection criteria are applied to variables that pertain to an event

as a whole, such as missing ET (E/T ). If an event fails to satisfy the requirements, the

event is thrown out. Jet selection criteria are requirements on individual jet candidates

which make use of jet variables. These are used to separate true hadronic jets from fake

jets caused by instrumentation problems. If a jet fails to satisfy the requirements, that

jet is not included in the count of jets in the event. For all remaining good events, the

jet multiplicity is the number of jets which pass the jet requirements. Sum of all the

\good" jet ET 's is called HT :

HT =Xjets

ET

Studies have been performed to determine the optimal requirements for each event and

jet variable. The results of these studies have been documented [27] [28] and are used as

a starting point in this analysis.

5.3.1 Criteria for missing ET

Recall that missing ET is a measure of \missing energy" in a system that is designed to

surround the interaction vertex and detect nearly all the �nal-state particles produced.

The presence of missing ET (> few GeV) indicates either the production of weakly-

interacting particles (such as a neutrino), or a mis-measurement of energy depositions in

the detector. For example, a cosmic ray passing through the detector during a collision

will deposit energy into the calorimeter. In most cases this results in an imbalance in

transverse energy and consequently a large missing ET relative to the leading jet(highest

ET jet) ET . We require, then, that the ratio of the event missing ET (E/T ) to the ET of

43

Figure 5.2: Logarithmic and linear plots of missing ET /E(1)T . We require this ratio to

be less than 0.7.

the leading jet be less than 0.7 ( Figure 5.2.) This requirement was designed to remove

events that coincide with cosmic rays passing through the calorimeter [27] [28].

5.3.2 Jet selection criteria

The following selection criteria are applied to individual jet candidates in an event. If

the requirement is met, the jet is retained and counted in the jet multiplicity of that

event.

Jet pseudorapidity (�)

An � requirement is applied which removes jets with an � greater than 3.5. The D�

calorimeter is instrumented to � = 4:2. A jet with its �nal determined direction very near

the boundary of � = 4:2 would likely have some of its energy in a region beyond � = 4:2.

The measured energy would then be an underestimate of its true energy. Because we

are using a cone size R of 0.7 in our jet �nding algorithm, we place a limit on jets with

a measured � < 3:5 so that even at that �, the full 0.7 cone is within the instrumented

region of the calorimeter.

44

Figure 5.3: Electromagnetic Fraction for all candidate jets. We require jets to have anEM fraction between 5% and 95%.

Electromagnetic fraction (EM)

The jet electromagnetic fraction (EM fraction) is the fraction of a jet's energy that is

deposited in the electromagnetic layers of the calorimeter. The requirement is that jets

have an EM fraction between 5% and 95% (Figure 5.3.) Very low values of the EM

fraction are due to calorimetric noise in the coarse and �ne hadronic layers, while very

high EM fraction values are due to EM calorimeter noise, electrons, or photons [28].

Table 5.2: Event and jet selection criteria (S.C.)

Name Description

Event S.C. Missing ET ET(1)=E/T = 0:7

Njets Njets � 2Jet � all jets have �3:5 < � < 3:5

Jet S.C. CH fraction CHF < 0:4EM fraction 0:05 < EMF < 0:95Hot Cell ratio Hottest/2nd hottest< 10:

45

Figure 5.4: Coarse hadronic fraction for all candidate jets. We ag as good all jets withCHF< 0:4.

Coarse hadronic fraction (CH)

This requirement is designed to remove fake jet candidates caused by activity in the Main

Ring. The Main Ring is the next-to-the-last stage in acceleration before the protons and

antiprotons are transferred to the Tevatron ring. At D�, the main ring passes through

the upper part of the calorimeters (both central and end) in the coarse hadronic section

(see Figure 3.9). When protons and antiprotons are accelerated in the main ring at the

same time Tevatron collisions are taking place, a collision in the main ring between the

proton bunches and gas molecules will result in showers that deposit energy primarily in

the hadronic section of the calorimeter. A jet candidate with a high fraction of its energy

in the coarse hadronic section is most likely a jet coming from beam-gas interactions in

the main ring. The limit placed on the CH fraction is 40%, meaning that if more than

40% of a jet's energy is in the coarse hadronic section of the calorimeter, the jet is

rejected as a main-ring e�ect (Figure 5.4).

46

Hot cell ratio

The energy of a jet is distributed over many calorimeter cells. When one cell in a cluster

Figure 5.5: Hot cell ratio distribution. This is the ratio of the energy of the mostenergetic cell to the second most energetic cell. This ratio must be less than 10 to agthe jet as good.

has much more energy than the other cells (a \hot cell"), it is likely that the extra energy

is due to instrumentation e�ects. The requirement for a good jet is that the energy in the

most energetic cell can be no more than 10 times the energy of the next-most-energetic

cell (see Figure 5.5).

5.4 E�ciencies of the inclusive jet triggers

Triggers used in �ltering the data stream from the detector may su�er ine�ciencies which

can bias the resulting data set. The e�ciency of a trigger is de�ned as the probability

that an interaction having the required properties actually passes the trigger criteria as

implemented. One source of ine�ciency is reduced accuracy in the quick measurements of

the event's characteristics (like jet ET ). This loss of accuracy can lead to false-positives

and false-negatives. The result of these errors in triggering is called trigger bias. These

47

errors are more pronounced when the event's characteristics are near the boundaries of

the trigger.

Studies have determined the e�ciencies of the single-jet triggers used in this analysis

[29] [30]. The e�ciencies are determined from data taken during collider runs called

\Mark-and-Pass" runs. In these runs, all events passing the �rst level (Level 0) trigger

are recorded, regardless of whether they pass the second and third trigger levels. The

comparison of events that would have failed to pass the trigger and those that pass

characterizes the e�ciency of the trigger. This e�ciency is measured as a function of

event characteristics such as leading jetET and leading jet direction (in �). The measured

e�ciencies are parametrized for use in a FORTRAN subroutine.

The average trigger e�ciencies as a function of HT (PET ) of the �ve triggers used

in this study are shown in Figures 5.6, 5.7, and 5.8. The plots show the average

Figure 5.6: Event trigger e�ciency vs HT for di�erent jet multiplicities (20 [A] & 30 [B]GeV triggers).

event trigger e�ciency as a function of the HT of the event for events with 2, 3, 4, or 5

48

Figure 5.7: Event trigger e�ciency vs HT for di�erent jet multiplicities (50 [A] & 85 [B]GeV triggers).

Figure 5.8: Event trigger e�ciency vs HT for di�erent jet multiplicities (115 GeV trig-ger).

49

jets. In the low HT region, high-multiplicity events have lower average trigger e�ciency

than low-multiplicity events. The di�erences in e�ciency decrease at higher HT . The

reason for this e�ect is that at a particular HT , a high-multiplicity event shares its total

transverse energy with more jets, reducing the average jet ET . This results in an overall

lower trigger e�ciency for high multiplicity events in HT ranges just above the trigger

threshold.

5.5 Measurement of the ratio of cross sections

As seen in Figure 5.1, each event contains anywhere from two to 9 jets. We calculate for

each event a quantity called HT de�ned as the sum of the jet transverse energies:

HT =Xjet i

E(i)T

for all jets above a �xed ET threshold. We produce a distribution of HT for all events.

In each HT bin we calculate the fraction of events that have three or more jets:

# events with 3 or more jets

# events with 2 or more jets

This ratio is equivalent to the cross section ratio�3+�2+

:

�3+�2+

=�( pp ! n jets +X ; n � 3)

�( pp ! m jets +X ; m � 2)

since factors relating the number distribution to the di�erential cross section cancel out in

the numerator and denominator. Figure 5.9 shows the measured ratio for all �ve triggers

over the �ll range in HT . The di�erent sets of points have similar characteristics. There

is an initial steep increase in the cross section ratio with HT , which levels o�. All �ve

curves from the �ve triggers level o� at roughly the same value (0.7). The sharp increase

occurs at di�erent regions of HT related to the trigger threshold. The reason for this is

50

Figure 5.9: �3+�2+

for all �ve triggers. The larger the trigger threshold, the higher the

HT value at which �3+�2+

\saturates." The vertical lines at the top indicate the triggerused in di�erent HT ranges.

51

Table 5.3: HT range used in each trigger.

Trigger HT range (GeV)JET 20 80-170JET 30 170-240JET 50 240-330JET 85 330-430JET 115 430-600

primarily kinematic but is also a�ected by the trigger e�ciency. In the case of the 50

GeV trigger (H), for example, a three-jet event requires at least (50+20+20)=90 GeV

to register in the cross section ratio, so the curve for that trigger has a starting point

of 90-100 GeV. In order to test this, we compare the results to simulations using the

Monte Carlo event generator HERWIG [31] [32]. The event generator is run using �ve

di�erent con�gurations corresponding to the �ve data triggers. In each con�guration, a

jet is required to have an ET greater than 20, 30, 50, 85, and 115 GeV. The ratio �3+�2+

is calculated for each, and compared to the measurement from the data. As seen in

Figure 5.10, the HERWIG result is very similar to the data, with a small di�erence in

overall normalization.

The \changeover" point from one data trigger to the next is chosen as the point

where the curves for the two triggers coincide within errors. The changeover points are

marked on the plots with vertical lines at the top, and Table 5.3 lists the HT range

used with each trigger. Figure 5.11 shows the cross section ratio using events from each

trigger in the optimizedHT ranges of Table 5.3. The measurement begins at an HT of 80

GeV. This point is chosen to be as low as possible but still have a high trigger e�ciency

of events in the region. The starting points for each trigger are also checked to ensure

that the events are also trigger e�cient. A correction for trigger ine�ciency is made by

scaling the event weights appropriately. In all cases the corrections amount to less than

1%.

52

Figure 5.10: �3+�2+

for all �ve pseudo-triggers, using HERWIGMonte Carlo. The triggeringe�ect was simulated by requiring the highest ET jet to have an ET above the chosenthresholds.

53

5.6 Minimum jet-ET requirement

The jet multiplicity of an event is the count of good jets above a minimumET threshold.

Changing this threshold changes the jet multiplicity as well as the HT of an event. The

lower bound of the jet ET threshold is determined by calorimeter response nonlinearities

[43] , uninstrumented regions in the calorimeter, trigger e�ciency, and reconstruction

jet-�nding e�ciency. In this analysis we vary the minimum jet ET from 20 GeV to 40

GeV and analyze the e�ect of changing the value of this threshold. Figure 5.12 shows

�3+�2+

vs HT for minimum ET thresholds of 20 and 30 GeV. The shape of the two curves

are similar, but, as expected, the ratio decreases with increasing minimum jet ET over

the full range in HT .

54

Figure 5.11: �3+�2+

showing the breakdown of data points from di�erent triggers.

55

Figure 5.12: �3+�2+

for two values of mininum jet ET .

56

Chapter 6

Sources of Measurement

Uncertainty

6.1 Introduction

In this chapter, I will list the uncertainties in the measurement of the cross section

ratio. The systematic errors are due to the use of jet and event selection criteria, the

possible mis-measurement of the interaction vertex, multiple-interaction events at high

luminosities, and the jet energy scale correction.

6.2 Trigger e�ciency correction

A correction for trigger ine�ciency is applied to the data based on the measured event

trigger e�ciencies. The uncertainty in the ratio �3+�2+

due to the trigger e�ciency cor-

rection is about 3% at HT = 80 GeV and quickly drops to zero beyond 150 GeV.

The magnitude of this error depends on the jet ET threshold used. The uncertainty is

also 1-3% for jet thresholds of 25 GeV, while at 30 and 40 GeV the error is negligible.

This trigger uncertainty is uncorrelated from point-to-point and is included in the total

systematic error.

57

Figure 6.1: Systematic error due to event and jet selection criteria.

6.3 Jet and event selection criteria

The systematic error in the cross section ratio measurement due to event and jet selec-

tion criteria is estimated by comparing the measured ratio before and after the individual

selection criteria are applied. The di�erence between the two measurements is a conser-

vative estimate of the error. The percentage change in the ratio as a function of HT from

each of the selection criteria is shown in Figure 6.1 and listed in Table 6.1.

58

Figure 6.2: Change in jet ET after re-vertexing. The di�erence is (re-vertex ET - regularET ). A jet's ET can increase or decrease with re-vertexing, and the small negative meanshows that on average a jet's ET will decrease with re-vertexing.

6.4 Vertex dependence

The interaction vertex for each event is determined by central drift chamber (CDC). The

CDC measures tracks made by charged particles passing through it. The reconstructed

tracks are extrapolated to the z-axis, resulting in one, two, or three groups of z-axis

intercepts. The intercept value which has the largest number of tracks pointing to it is

called the primary vertex. The others are called secondary, etc. vertices [33]. An accurate

measurement of the interaction vertex is important because the transverse energy and

pseudorapidity of a jet are calculated using the measured interaction vertex. If the wrong

vertex is chosen as the primary vertex then the measurement of jet ET and � and thereby

HT and�3+�2+

are a�ected.

To estimate the e�ect of choosing the wrong vertex we calculate the vertex based on

an estimate of the missing ET (E/T ). The E/T is calculated using both the primary and

secondary vertices. The vertex which produces a smallerE/T is presumed to be the correct

one. Using this new vertex, the jet ET and pseudorapidities are recalculated. About 16%

of events are \re-vertexed" in this manner. A characteristic of this re-vertexing is that

jet ET s are reduced a little on average. The distribution of jet ET s in those events

that were re-vertexed is shown in Figure 6.2. The change in the inclusive-two-jet and

inclusive-three-jet HT distribution is shown in Figure 6.3. The changes in the two-jet

59

Figure 6.3: Change in two-jet inclusive (A) and three-jet inclusive (B) HT from re-vertexing.

Figure 6.4: Change in �3+�2+

from re-vertexing.

60

Figure 6.5: A. Single interaction fraction vs instantaneous luminosity. B. Instantaneousluminosity distribution.

inclusive HT distribution and three-jet-inclusive HT distribution cancel when the ratio is

taken. Figure 6.4 shows the change in the cross section ratio following the re-vertexing

correction. The error in the ratio �3+�2+

is correlated from point to point and we therefore

do not include it in the total systematic error.

6.5 Luminosity dependence

Events were recorded during widely-ranging luminosity conditions. The higher the lu-

minosity the greater the average number of pp interactions per bunch crossing. That is,

for any pp interaction (event) that triggers the �rst-level (Level 0) detectors, the chance

of a second pp interaction in the same bunch crossing increases with the luminosity of

the beam (Figure 6.5). The second interaction is typically a di�ractive (glancing) inter-

action which results in the production of particles that deposit energy into the forward

calorimeters (at large j�j) close to the beamline. These energy deposits typically do not

61

cluster together and mimic jets, but the measurement of a jet from the primary hard

interaction is contaminated by this energy deposited from the second interaction. This

is partially corrected for in the latest version (V5.1) of the jet energy scale, described in

Section 6.6 [34].

The result of mis-measured energy due to multiple interactions is a jet ET (and hence

event HT ) that is higher than the true ET or HT . This causes the �3+�2+

vs HT curve to

uctuate downward. Another e�ect of jet ET s uctuating upward is that jets that in

reality have less than the 20 GeV minimum ET requirement will then have a measured

ET greater than 20 GeV and will thus be counted as a jet. The result of this thresh-

old e�ect is a change in the jet multiplicity which may change the cross section ratio

measurement when the jet count change is from 2 to 3 jets.

To measure the luminosity dependence, we measured the cross section ratio as a

function of instantaneous luminosity for several ranges in HT . This range has to be

chosen carefully because the cross section ratio increases rapidly with HT at low HT .

Figure 6.6 shows the measurement of �3+�2+

as a function of instantaneous luminosity for

a number of bins in HT . There is no consistent pattern of increase or decrease in the

cross section ratio with the luminosity. To estimate an error, we �t a straight line

through points and extrapolate the line to zero-luminosity. The di�erence between the

zero luminosity value of �3+�2+

and its value for all the luminosities is our estimate of the

uncertainty due to luminosity, about 1%.

6.6 Jet energy scale

The D� calorimeter is calibrated by measuring the response of calorimeter test modules

to pion beams of known energy (\test beam data"). The gives us a set of calibration

constants (relative sampling weights, conversion factors for ADC counts to GeV) that

are used to convert the instrument signals to a value with dimensions of energy. The raw

energy value is then further corrected for response, uranium noise, energy not coming

from the hard interaction, and algorithm and calorimeter resolution e�ects [35].

62

Figure 6.6: �3+�2+

vs Instantaneous luminosity (in 1030s�1cm�2) for di�erent HT ranges.

The relation between the measured raw jet energy and the jet energy is given by the

relation

Ejetparticle =

Ejetmeas �O(�R; �;L)

R(�R; �; E) [ 1� S(�R; �; E) ]

where O is an o�set function of the cone size �R, the pseudorapidity � of the jet, and

the instantaneous luminosity L. This function accounts for uranium noise, \pileup"

(distortion of calorimeter signal due to \out-of-time" signals caused by the memory

of the electronics [36]) , and multiple interactions. It also takes into account energy

63

Underlying event

High-pT jet

p

p

Down the beampipe

Perpendicular to beam axisHigh-pT jet

Perpendicular to beam axis

Underlying eventDown the beampipe

Figure 6.7: Underlying event

deposition from the underlying event, i.e., interactions involving quarks from the proton

and antiproton that do not undergo hard scattering (Figure 6.7).

R is the response function of the calorimeter to incident particles as a function of

the particle energy. (Figure 6.8). This removes e�ects from jets that deposit energy into

uninstrumented regions between calorimeter modules. It also corrects for an e=� ratio ( a

measure of the relative response of the detector to electrons and hadrons) which deviates

from the ideal of unity. An e=� ratio of 1 is needed for a particle-independent correction.

[37]. A at response means that a calorimeter cell's output signal scales linearly with

the energy of the incident particle. In Figure 6.8 this would correspond to a horizontal

line. Response is determined by �rst measuring the electromagnetic (EM) scale using

dielectron and diphoton decays of known particle resonances (Z, J= , neutral pions).

Then direct photon + jet events are used to relate the EM scale to the hadronic scale.

S is a measure of how much energy is \lost" and unmeasured due to being beyond

the calorimeter jet algorithm cone (\out-of-cone showering"). In order for its energy to

be measured, a jet has to interact with and deposit energy into the detector. In the

process of interaction the particles that form a jet will produce a shower which may

extend outside the jet algorithm cone of R =p��2 + ��2 = 0:7. This will result in an

underestimate of the jet's energy. One method used to estimate the showering correction

is to compare jets from a Monte Carlo event generator with test-beam data of jets.

64

Figure 6.8: Jet energy scale response as a function of jet energy.

Our study of the energy scale error will concentrate on the response function R, the

uncertainty of which re ects the combination of all these e�ects. The nominal response

(solid curve) shown in Figure 6.8 is a �t to the \calorimeter response vs incident particle

energy" measurements, while the upper and lower envelope is the maximum and mini-

mum of all other possible �ts. These upper and lower response curves are extremes of

the energy scale. In addition, two alternate response curves (Figure 6.9) are also used

which intersect each other. The curve labelled `1' has a lower-than-nominal response at

low jet energy and a higher-than-nominal response at high jet energy. The second curve

has the opposite behavior: high response at low energy and low response at high energy.

Using these response curves we can check the e�ect of extreme variations in the energy

scale on the measurement of �3+�2+

vs HT .

These response curves are used to scale the jet energy. The di�erent measures of the

cross section ratio from these �ve curves are used to estimate the overall energy scale

error in �3+�2+

as shown in Figure 6.11d. The decreasing error from 80-300 GeV is due to

the upper and lower response curves, while the at region above 300 GeV is due to the

crossed response curves. The plot in Figure 6.10 shows �3+�2+

vs HT for the nominal, high,

and low values of the energy scale correction and the percentage error determined by the

65

Figure 6.9: Jet energy scale response (crossed) as a function of jet energy.

fractional di�erence between nominal and high, and low and nominal. The uncertainty

in the ratio �3+�2+

due to the energy scale correction is estimated by comparing the ratio

calculated from data corrected with the upper and lower response curves, as well as

the crossed response curves, and comparing to a calculation using the nominal response

curve. The error is set at 3%, correlated from one HT bin to the next.

6.7 Error summary

6.7.1 Statistical errors

The measurement of the ratio �3+�2+

involves a requirement on the jet multiplicity and a

count of the number of events passing the multiplicity requirement. Thus, the statistical

error on the ratio is given by the binomial formula. If the original sample has B events

and the multiplicity requirement reduces that number to A then the statistical error is:

error =

qA(1� A

B )

BA < B; B 6= 0

The HT range is binned so that the statistical error is less than 10%.

66

Figure 6.10: �3+�2+

vs HT for di�erent energy scale choices.

67

Table 6.1: Errors due to di�erent sources

Source Error (percent)Trigger 1-3 % HT < 150 GeVCH Fraction 0.2 %EM Fraction 0.3 %Hot Cell Ratio 1-1.5 %Missing ET 0.5 %Mis-vertexing 0.7%Luminosity 1%Jet Energy Scale 3%Total Systematic Error 3-4 %

6.7.2 Total systematic error

Uncertainty due to trigger, jet selection, and the statistical error are all point-to-point

uncorrelated and added in quadrature. The energy scale error is correlated and is handled

using a covariance matrix method (see Appendix B). The luminosity and mis-vertexing

error are negligible and are not included. The uncertainties as a function of HT are

shown in Figure 6.11. The sum of all uncorrelated errors is shown in the lower right

plot. The errors are listed in Table 6.1.

6.7.3 Correlated errors

In comparing our measured cross section ratio with theoretical predictions we wish to

correctly include systematic errors that are point-to-point correlated, that is, the er-

rors that shift all points in the same direction (up or down). Including these errors as

uncorrelated would overestimate the agreement between data and theory. Figure 6.12

shows the cross section ratio with statistical and uncorrelated systematic errors (added

in quadrature) on the points, while the magnitude of the correlated systematic errors is

represented by a shaded band at the bottom of the plot.

68

Figure 6.11: Systematic errors. The bottom two plots show the total correlated anduncorrelated errors.

69

Figure 6.12: Cross section ratio showing statistical and systematic errors.

70

Chapter 7

Monte Carlo Simulations of QCD

Processes

7.1 Introduction

The measurements of the ratio �3+�2+

can be compared to theoretical calculations in the

form of a Monte Carlo event generator. An event generator is a program that simulates

a physical process . In our case, the process is the pp collision at Fermilab. The \inputs"

to the event generator are the operating parameters such as total center-of-mass energy.

The output is a set of events that are representative of events taken with a detector.

This chapter will brie y discuss the Monte Carlo method as applied to a matrix-element

cross section calculation program (JETRAD) and the modi�cations implemented to test

the soft-jet renormalization scale prescription. We then present the results of the com-

parisons between the data and theory.

71

7.2 Monte Carlo prescription

A Monte Carlo method is a method for evaluating di�cult integrals or of sampling

probability distributions. The basic theorem of Monte Carlo integration is that the

integral of a function f over a multidimensional volume V can be estimated by taking

the arithmetic mean of the function f over many points N sampled from the volume V .

ZfdV � V hfi with hfi � 1

N

NXi=1

f(xi)

The points xi are sampled from the volume V using a pseudo-random number generator.

7.3 JETRAD

JETRAD [38] is a next to leading order Monte Carlo event generator for inclusive 1 or

2-jet production for pp or pp collisions. One chooses the type of collision (here pp ), the

center of mass energy, the number of jets to produce, the order in �S of the perturbative

calculation, the pseudorapidity range for �nal state partons, and the renormalization and

factorization scale. In addition to calculating the total cross section for the interaction

speci�ed, it can also produce distributions of event variables. While the program only

generates two- and three-jet events, the two-jet and three-jet cross sections are inclusive

quantities, i.e., they are the cross section for two-or-more and three-or-more jets [39].

7.3.1 General approach

We begin with the factorization theorem of perturbative QCD

� =

Zdx1dx2d�

1

2sfh!i(x1)fh!j(x2)

XjMj2

where dx1 and dx2 are the momentum fractions of the partons from the proton and

antiproton, d� is the Lorentz-invariant phase space element, s is the ux factor for the

partonic cross section, f(x) are the parton distribution functions, andPjMj2 is the

72

squared matrix element for the partonic cross section, summed over �nal state colors

and polarizations, averaged over initial colors and polarizations.

For n particles in the �nal state, the integration above is (3n � 2)-dimensional. It

is this integration that will be done via Monte Carlo. We �rst rewrite the di�erential

element as

1

2sdx1dx2d�n = J

3n�2Yi=1

dri

where J is the Jacobian of the transformation. The Monte Carlo prescription then

approximates the integral

� =

Zdr1 � � �dr3n�2 J f(x1)f(x2)

XjMj2

with the sum

� � 1

N

Xfrig

Jf(x1)f(x2)X

jMj2

where frig are N sets of 3n � 2 random numbers [40]. In the case of the JETRAD

event generator, the numerical integration routine used is VEGAS [41]. It supplies the

frig random numbers and the \weight factor" w = 1=N . Once can produce arbitrary

distributions d�=dz by �lling a histogram of z with the weight

Jf(x1)f(x2)PjMj2

N�z

where �z is the width of the bins in z.

To produce a calculation of �3+�2+

vs HT , one produces two distributions d�3+=dHT

and d�2+=dHT and divides the �rst by the second. A plot of this is shown in Figure 7.1.

73

Figure 7.1: JETRAD calculation of �3+�2+

vs HT .

74

7.3.2 Input parameters and selection criteria

Introduction

The input parameters in JETRAD include the kinds of particles to be scattered (pro-

ton or antiproton), the center-of-mass energy of the collision, the number of loops in

the calculation, the number of jets to produce (one, two, or three), the minimum and

maximum ET of the jets, the � ranges available for the jets, the \cone size" of the jet

cone used in the jet-�nding algorithm, the parton distribution function set to use, the

renormalization scale (maximum jet ET in the event or the sum of all jet ET s in the

event), and the number of events to generate. The factorization scale is set equal to the

renormalization scale.

Cone size and Rsep

The jet-�nding algorithm used in the Monte Carlo event generator approximates the

algorithm used in the data reconstruction. The data jet algorithm searches for a jet by

looking at the way energy is deposited into calorimeter cell towers. In JETRAD, the

products of the interaction are two or three partons. In the data, reconstructed jets that

are very close or overlapping are either combined or separated. A similar mechanism

is used for the Monte Carlo jets, using the Rsep parameter. The Rsep is de�ned as the

maximum angular distance, divided by the cone size, allowed between two partons to

be merged into a single jet. Rsep is used to approximate the split-merge decision in the

data version of the jet-�nding algorithm. See Figure 7.2. To determine the proper

value of Rsep for the data, the following method was used: Take a single jet from one

event and place it (cell-by-cell) into an independent event. The second event was then

re-reconstructed and the number of jets found is counted. Also, the distance (D) from

the inserted jet to the closest jet in the independent event was calculated. The change

in the number of jets found with D was then plotted (Figure 7.3) [25]. In the �gure, the

value of D where half of the events have two reconstructed jets is 0.85. This translates

75

∆ R

∆

Combine partons into 1 jet

R

jet 2

jet 1

calorimetercells

parton 1

∆

Partons considered separate jets

R = 1.10 > Rsep x cone-size∆R = 0.90 < Rsep x cone-size

parton 2

parton 1parton 2

1.1

0.90

Figure 7.2: A. Splitting and merging of jets in the jet reconstruction algorithm. Theshaded rectangles represent calorimeter cells that are shared by two jets before the split-merge criteria is applied. B. Use of Rsep in Monte Carlo jets. For a cone-size of 0.7 andan Rsep of 1.3, the critical distance for merging two partons into a jet is 0.91.

to a value of Rsep of DR=conesize = 0:85=0:7 = 1:2. An alternate method of measuring

Rsep based on jet shape variables (e.g. the average fraction of calorimeter cell ET in

a subcone of a jet) gives a range for Rsep of 1.2 to 1.4. From these two methods an

Rsep value of 1.3 was chosen. This value is used in the jet �nding algorithm that is

implemented in JETRAD. When two partons have an angular distance of Rsep �cone-size = 1:3�0:7 = 0:91 the two partons are merged into one jet. The choice of Rsep a�ects

the fraction of jets merged for certain critical values of D. To estimate the uncertainty

due to the choice of Rsep , we take the width of the transition region in Figure 7.3 and

use that as the range over which we vary Rsep . The D range is 0.8 to 0.933, which

translates to 0.2 units in Rsep . Thus we vary our chosen Rsep of 1.3 by �0:1 units. Thechange in the three-jet HT distribution and in �3+

�2+is the estimate of the uncertainty.

The value chosen for Rsep can a�ect the number of jets reconstructed in the Monte

Carlo and hence the value of �3+�2+

. We compare the choice of Rsep = 1:3 (standard) to

Rsep values of 1.2 and 1.4. This range corresponds to the region in Figure 7.3 where the

number of jets is between one and two. We then compare the change in the three-jet

HT distribution and also in �3+�2+

for both regular and smeared (Section 7.3.3) jets using

76

Figure 7.3: Di�erence in the number of reconstructed jets vs D for a cone of radius 0.7The input jet has ET � 20 GeV and j�j � 4:0.

Figure 7.4: The fractional di�erence between Rsep =1.3 and 1.2 for the three-jetHT distribution (A) and the ratio �3+

�2+(B).

77

Figure 7.5: The percentage change in HT and�3+�2+

as we increase or decrease Rsep . Asmaller Rsep will cause less merging of partons and thus more jets. A larger Rsep willproduce more merging and less jets.

all three values of Rsep . Figure 7.4 shows the change in the 3-jet HT distribution and

the cross section ratio between an Rsep of 1.3 and 1.2. (The di�erence with Rsep =1.4 is

similar in shape and magnitude.)

The percentage di�erence varies slightly with HT but is more or less constant at

around 3%. The plot in Figure 7.5 shows the percentage change in three-jet HT and�3+�2+

as we move away from an Rsep of 1.3 in either direction. By de�nition the change at 1.3

is zero. Both the three-jet HT distribution change and the cross section ratio change

have the same Rsep sensitivity. The sensitivity of the theoretical cross section ratio to

the choice of Rsep is around 3%.

Parton distribution

It is predicted that since we are measuring a ratio of cross sections, the e�ect of using

di�erent parton distribution functions is minimal. This was studied empirically using

two di�erent parton distribution sets CTEQ4M and MRSD00. The cross section ratio

78

Figure 7.6: �3+�2+

from JETRAD using A) CTEQ4M and B) MRSD00 parton distribution

functions. C) shows the percentage di�erence, which uctuates about zero.

79

�3+�2+

calculated with each of these parton distribution sets is shown in Figure 7.6. The

di�erence between the two parton distributions sets is negligible.

7.3.3 Sources of theoretical uncertainty

Jet ET resolution

The jet ET resolution was studied with events containing two back-to-back jets, using

momentum conservation in the transverse plane de�ned by the two jets [42] [43]. The

calorimeter resolution for measuring jets was modeled after the single particle resolution,

i.e., the calorimeter resolution for measuring single incident particles. The single particle

calorimeter resolution is parametrized as

��ET

ET

�2

= C2 + S2=ET +N2=ET2

The C is an constant o�set, the S2=ET term is from contributions from sampling uc-

tuations and the nature of the incident particle, and lastly the N2=ET2 term is due to

noise uctuations a�ecting the high energy regime [43].

The measured resolutions are shown in Figure 7.7 for various jet � regions. The

measurement of jet ET generally improves as both ET and � increases. In order to

compare the data to the Monte Carlo prediction, Monte Carlo jets are \smeared" using

the measured jet resolutions. The smearing involves adding or subtracting a random

fraction of the jet's resolution uncertainty to the Monte Carlo jet ET using the resolution

parametrizations. This changes the characteristic of the sample of Monte Carlo jets to

model the jets measured by the calorimeter.

The result of smearing the Monte Carlo jet ET s should not change the shape of the

jet pseudorapidity distribution (Figure 7.8). We compare the jet � distribution with

and without smearing, and �nd that the fractional percentage di�erence between the

smeared and raw � distribution uctuates about zero in the central region (� between

�1 and +1) and increases to about 10% at � of 3.5 as shown in Figure 7.9.

80

Figure 7.7: Run 1a jet ET fractional resolution. Dashed lines represent the upper andlower error estimates on the fractional resolution.

The change in shape at high � is accounted for by the tendency of high-� jets to have

a lower average jet ET , as shown in Figure 7.10. The curved line shows the limit for

jets produced from 1800 GeV proton-antiproton collisions. There can be no jet with an

ET above the curve for such large jet � s. The low-ET , high-� jets are then smeared

with a larger uncertainty value than central (0:8 < j�j < 1:6) jets.

We apply the smearing to the Monte Carlo jets, measure the ratio �3+�2+

and compare

to a sample that was not smeared. Figure 7.11 shows the change in the ratio when

the smearing is applied. The smearing is found to a�ect the shape of the ratio mostly

81

Figure 7.8: Jet pseudorapidity (�) distribution for JETRAD Monte Carlo jets. Thehistograms are normalized to the same area.

Figure 7.9: Fractional percentage di�erence between smeared and raw jet pseudorapiditydistributions.

82

Figure 7.10: Average jet ET vs jet pseudorapidity (�). The curve shows the kinematiclimit for jets from 1800 GeV collisions. The small rise in the average ET in the � = 1:4region is due to di�erent calorimeter sampling weights in that pseudorapidity region.

Figure 7.11: Comparison of the ratio �3+�2+

calculated with JETRAD, with and withoutthe smearing of jet ET . The bottom plot shows the percentage di�erence between thetwo.

83

at the low-HT region. The smeared-jet �3+�2+

has a higher value than the measurement

without smearing. This is due to the nature of the jet ET spectrum: the steeply falling

distribution causes jets to \smear high" (smearedET larger than originalET ) more often

than it \smears low". This promotes jets whose ET is just below the threshold to above

the threshold, thus increasing the jet multiplicity. This results in a higher measured

cross section ratio.

The measured resolutions shown in Figure 7.7 include an uncertainty envelope indi-

cated by the dotted lines. We use this resolution uncertainty to estimate the uncertainty

in �3+�2+

due to the application of the smearing procedure. The method is as follows: each

fractional resolution �ET=ET vs ET is accompanied by two other curves (dashed). We

call these the \Hi-smear" and \Lo-smear" resolution curves. We use each of the Hi- and

Lo-smear curves to smear the Monte Carlo jets and compare to the smearing with the

nominal curve. The fractional di�erence in the ratio �3+�2+

between the nominal, Hi and

Lo smear is shown in Figure 7.12. We estimate the uncertainty in the ratio �3+�2+

due to

Figure 7.12: The fractional di�erence between the nominal smear and the Hi and Losmearing.

84

the smearing procedure to be 2% constant in HT . This is taken to be correlated from

one HT bin to the next.

Jet-�nding e�ciency

The jet reconstruction algorithm in RECO has an e�ciency for �nding jets as well as

an e�ciency for reconstructing the jet's correct energy. Any ine�ciency which results

in RECO calculating the wrong energy of a jet is corrected for in the jet energy scale.

There is, however, the possibility of RECO not reconstructing a jet at all ( missing a jet

altogether). This has been studied [44] [45] using photon+jet events. The data set used

consists of single-photon events where a balancing jet is presumed to have been missed

by the jet reconstruction algorithm. The result of the study is a set of e�ciency curves

for di�erent � regions. In each � region, the e�ciency for RECO to �nd (or not miss)

a jet is given as a function of jet ET , as shown in Figure 7.13. Jet-�nding e�ciency is

always greater than 95% for ET above 20 GeV and rises quickly to 100% above 25 GeV.

Using this information, we can either correct the data and scale up the jet count

for some events, or correct the Monte Carlo and scale down the jet count appropriately.

We choose the latter method, randomly dropping jets in the Monte Carlo based on the

jet-�nding e�ciency of Figure 7.13 to simulate the mis-reconstruction of jets in the data.

The change in the cross section ratio as a result of applying this e�ciency is shown in

Figure 7.14. Since this correction is so small it is not included in the �nal results.

Total error

The systematic error due to jets not reconstructed and jet resolution is added to the

statistical error in quadrature. This is shown in Figure 7.15. The total error is between

2 and 2.5% .

85

Figure 7.13: Jet-�nding e�ciency in di�erent pseudorapidity regions as a function ofjet ET .

Figure 7.14: Percentage error in �3+�2+

as a function of HT due to jets not reconstructed.

86

Figure 7.15: Errors in the theoretical prediction. The horizontal axis is HT .

7.3.4 Renormalization scale of third-jet production

In calculating a cross section for a process, one �rst chooses a renormalization scale

�R. The Standard Model does not give a prescription for choosing this scale. It is

typically chosen to be of the order of the transverse energies of the jets produced. The

renormalization scale �R is a parameter in the approximation of �S .

�S =�S(�

20)

1 + (�1=4�)�S(�20) ln(�2=�20)

(lowest order)

where �S(�20) is a boundary condition for the di�erential equation whose solution is �S

above. �1 is 11� 2nf=3 with nf quark avors.

In the current version of JETRAD the same renormalization scale is used in eval-

uating �S for both the hard scatter and also the gluon emission leading to a third jet.

87

The choice for this scale is typically a factor times HT (=PET ). We are interested

in testing whether a di�erent renormalization scale for the third jet production is more

consistent with experimental �ndings than is the use of one renormalization scale for all

instances of �S .

The two di�erent theoretical parametrizations are shown in Table 7.3.4. In the �rst

instance �(3)R � HT , i.e. the renormalization scale for both the hard scatter and third-jet

emission is the same. This is the standard JETRAD prescription. In the second case

�(3)R � E

(3)T , i.e. the third-jet scale is e�ectively changed to the value of the ET of the

third jet (when there is a third jet). This is done by scaling the cross section by a factor

�S(�R = E(3)T )

�S(�R = 14HT )

where �R = E(3)T is a renormalization scale on the order of the transverse energy of the

third jet [46]. The di�erence between the two theories above is in the third jet. We

calculate �3+�2+

vs.HT with both choices of soft-jet renormalization scales and compare to

data. Figure 7.16 shows the theory calculation using two di�erent renormalization scales

for the third jet emission. In order to compare the relative magnitudes of the third jet

ET with HT the average ET of the third jet is plotted as a function of HT (Figure 7.17).

Table 7.1: The scales for the hard scatter and third jet emission for two JETRADsettings.

Theory Hard Scatter Scale Third jet scale

�(3)R � HT �R = 1

4HT �(3)R = 1

4HT

�(3)R � E

(3)T �R = 1

4HT �(3)R = E

(3)T

88

Figure 7.16: JETRAD calculation of �3+�2+

using two di�erent renormalization scales forsoft jet emission.

7.4 Comparison of data with QCD

7.4.1 Introduction

In the next sections we compare the two renormalization scale prescriptions to the data

to determine whether one o�ers a more consistent description of the measurement than

the other. The two theory candidates being compared di�er only in the choice of the

Figure 7.17: The average value of the third jet ET as a function of the HT of an event.

89

renormalization scale for the third jet (�(3)R ). The choices are �

(3)R = 1

4HT and �(3)R = E

(3)T .

In order to compare the data to the theory, the latter is rebinned to match the binning

of the data. The �2 for the data-theory comparison is given by:

�2 =X

HT bins i j

(Di � Ti)1

Cij(Dj � Tj)

where Di is the value of�3+�2+

in the ith HT bin for data and Ti is for the JETRAD (The-

ory), and Cij is the covariance matrix containing all the data and theory uncertainties as

well as correlations between one HT bin and another. (See Appendix B for an example.)

A �2 per degree of freedom (or reduced �2, e�2) of about 1 indicates good agreement

between the data and theory. Comparing the �(3)R � HT case and �

(3)R � E

(3)T theory

curves to the data we have the result shown in Figure 7.18. Between the two theories,

Figure 7.18: A �2 comparison of data with two theory prescriptions.

the �(3)R � HT set is a better �t to the data, with a e�2 of 1.02 vs 6.8 for �(3)R � E(3)T .

Variation with the hard scale

The hard scale chosen in the comparisons of the previous section was � = 14HT .

The 14 factor is an arbitrary value but has shown to be in agreement with other D�

90

Figure 7.19: �2 di�erence between data and theory for di�erent hard scales. The pointsfrom the initial � = 1

4HT comparison are circled.

analyses which use JETRAD as the theory prediction, such as the single-inclusive jet

ET measurement. We have looked at the �2 di�erence between data and theory for hard

scales other than 14HT . Other factors we investigated are 0.2, 0.27, 0.3, 0.35, 0.4, 0.45,

and 0.5. We will use the variable � to denote the factor of HT used as the scale for the

hard scatter. The change in e�2 as we increase and decrease � from 0.25 is shown in

Figure 7.19. The initial 14 factor result is marked with a circle in the plot. As seen in the

�gure, when varying � up and down the initial 14 value, the e�2 for �(3)R � E

(3)T decreases

with � while for �(3)R � HT the trend is a slow rise with �. The e�2 has a minimum at

� = 0:25.

91

Variation with the soft scale

The choice of E(3)T as the soft scale is also somewhat arbitrary. To examine the other

possibilities we tried 1:5�E(3)T and 2:0�E(3)

T . Changing the coe�cient of the third-jet

ET from 1.0 to 2.0 changes mostly the normalization of the cross section ratio, with

a small change in the shape, as shown in Figure 7.20. We compare each of the three

Figure 7.20: The cross section ratio for three di�erent soft-scale choices. The thickshaded line is the cross section ratio using HT as the soft scale.

soft scale choices with the data and calculate a e�2 as before. The results are shown in

Figure 7.21. Varying the factor of E(3)T for the third jet scale does not appreciably change

92

Figure 7.21: �2 di�erence between data and theory as a function of the coe�cient ofthe hard scale.

93

Figure 7.22: �2 di�erence between data and theory as a function of the coe�cient ofthe hard scale (�), for jet ET thresholds of 20 (a), 25 (b), 30 (c), and 40 GeV (d).

the agreement between data and theory, and all �(3)R � E

(3)T scales are an unlikely �t to

the data.

Jet threshold

In the previous section, a jet ET threshold of 20 GeV was used in the data and theory. We

have studied how the results depend on this minimum jet ET threshold. We reanalyze

the data using minimum jet ET s of 25, 30, and 40 GeV and repeat the �2 comparisons

with theory.

Figure 7.22 shows the e�2 vs � plot for jet ET thresholds of 20, 25, 30, and 40 GeV.

At a jet ET threshold of 25 GeV, the �(3)R � E

(3)T curves improve their agreement with

data, with the greatest change occurring at � < 0:4. The �(3)R � HT theory still give

94

better agreement with the data, with perhaps a shift in the � value for the minimum e�2.As we increase the threshold to 30 GeV, the four curves remain close together and their

e�2 decrease to about 1 for all values of �. There is no preferred scale in this comparison.

The result is the same when we require all jets to have an ET above 40 GeV. The e�2vs � curves all converge to about 1 for most values of �. The greatest sensitivity in

distinguishing between a �(3)R � HT theory and a �

(3)R � E

(3)T theory is then found when

using a jet ET threshold of 20 GeV or at most 25 GeV.

7.4.2 Restricted range of jet pseudorapidity

We investigated the applicability of the previous �ndings as we restrict the pseudorapidity

of the jets to 2.0 instead of 3.5 as before. We are interested in how our �nding change

when we explore a narrower region of the � phase space. We repeat this using only 20

GeV as our jet ET threshold since this is the ET threshold where we are most sensitive.

Figure 7.23 shows the change in e�2 as a function of �. Agreement is poor for all �(3)R �

E(3)T , with weaker dependence on �. For �

(3)R � HT , the agreement between data and

theory is very good, with a minimum e�2 at � = 0:35. As with the full � < 3:5 case, the

�(3)R � HT theory is a better �t to the data.

95

Figure 7.23: �2 di�erence between data and theory, vs the hard scale, � < 2:0

96

Chapter 8

Conclusion

We measured the cross section ratio �3+�2+

as a function of the summed transverse energies

of all jets in an event. The analysis was based on single-jet inclusive triggers. The

uncertainties in the result were due to the jet energy scale, background from multiple-

interaction, and general event and jet selection criteria. Our measurement provides

an estimate of the rate of soft jet production at di�erent parton-parton center-of-mass

energies. We �nd that, above 200 GeV in HT , the probability for emitting a third jet

is around 70%. This result will help in the design of triggers for detecting the Higgs

particle at the Large Hadron Collider, especially in high-luminosity environments.

Our measurement is compared to theoretical predictions from a Next-to-Leading

Order QCD Monte Carlo event generator (JETRAD). Uncertainties in the theoretical

predictions are mainly due to the exibility in the parameters used for jet reconstruction.

Comparisons of data to theory lead to the conclusion that

1. The ratio �3+�2+

is in good agreement with perturbative QCD predictions for multijet

production above an ET threshold of 30 GeV.

2. The renormalization scale for the creation of soft jets is better modeled using a

scale of the order of the hard-scattering scale (HT ) rather than the transverse

energy of the third jet.

97

3. This �nding also holds when we restrict the pseudorapidity of jets from j�j < 3:5

to j�j < 2:0

4. As we increase the jet ET threshold, the discrimination between the scales �(3)R �

HT and �(3)R � E

(3)T is decreased, notably for Emin

T > 30 GeV.

The model that best describes the data is one that uses only a single scale for both

the hard scatter and for the emission of soft jets. This simpli�es theoretical calculations

since the additional scale does not improve the prediction. Although the single scale,

in particular a value of about 0:25HT , has been shown to hold for only this process

(three-jet events), the present technique can be tested on other topologies.

Calculations of the matrix elements for purely gluonic three-jet processes at Next-

to-Leading Order (O(�3S)) are already available. When the full NLO calculations are

completed, a measurement of �S can be extracted from the measured ratio �3+�2+

of cross

sections.

98

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102

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103

Appendix A

Matrix Elements for 2-to-3

Parton Scattering

In Chapter 2 I listed the matrix elements for 2-to-2 scattering and also one of the formulas

for 2-to-2 scattering. Here I list the remainder of the formulas. The four general kinds

of processes are:

(A) q(p1) + q0(p2) ! q(p3) + q0(p4) + g(k)

(B) q(p1) + q(p2) ! q(p3) + q(p4) + g(k)

(C) q(p1) + q(p2) ! g(p3) + g(p4) + g(p5)

(D) g(p1) + g(p2) ! g(p3) + g(p4) + g(p5)

The matrix element for process A is given in Chapter 2. For process B, the equation is

X���M(B)���2 =

4g6

9

" s2 + s02 + u2 + u02

2tt0

!�8

3([14] + [23]) +

1

3[12; 34]

�

+

s2 + s02 + u2 + u02

2uu0

!�8

3([13] + [24]) +

1

3[12; 34]

�

� 2

3

(s2 + s02)(ss0 � tt0 � uu0)

4tt0uu0

�8

3([12] + [34]) +

1

3[12; 34]

��

104

For Processes C and D we introduce the notation

fijg � pi _pj

The matrix element for process C is

X���M(C)���2 =

2g6

81

3Xi=1

f1igf2ig(f1ig2+ f2ig2)f13gf14gf15gf23gf24gf25g

�"10f12g � 9

XP

f13gf24g+ f14gf23gf34g

+81

f12gXP

f15gf25g(f13gf24g+ f14gf23g)f45gf53g

#

The sums over P are over cyclic permutations of the momentum labels (3,4,5) of the

�nal state gluons. And �nally, for process D we have

X���M(D)���2 =

9g6

640

"XP

f12g4#"X

P

f12gf23gf34gf45gf51g#

�0@Y

i<j

fijg1A�1

where the sums are over the permutations of the �ve gluon labels.

105

Appendix B

Calculation of �2 Using a

Covariance Matrix

Here we give an example of calculating �2 using a covariance matrix. This method was

used in Chapter 7 to compare data to Monte Carlo theoretical predictions. For simplicity

we have three data points compared to three theory points. Figure B.1 shows the data

points and theory curve. The error bars shown on the data points are the statistical

(inner) and systematic error (outer). The data and theory numbers with errors are

given in Table B. The error matrix for the data errors is constructed as follows: The

Table B.1: Data for sample �2 calculation.

Data TheoryX-value Value (D) Stat. error �stat Syst. error �syst Value (T ) Error

(uncorrelated) (correlated) (uncorrelated)

1 5.20 0.16 0.26 5.00 0.102 5.90 0.18 0.30 6.00 0.122 7.10 0.21 0.35 7.00 0.14

uncorrelated data statistical errors (0:16; 0:18; 0:21) are squared and form the diagonal

106

Figure B.1: Sample data and theory values to describe the covariance matrix �2 cal-culation. The gray lines above and below the theory line represent the error on thetheory.

elements of the error matrix Edatauncorr:

Edatauncorr =

0BBB@�2stat;1 0 0

0 �2stat;2 0

0 0 �2stat;3

1CCCA =

0BBB@0:025 0 0

0 0:031 0

0 0 0:045

1CCCAThe correlated data systematic errors are put into a matrix which includes correlations

from one data point to another:

Edatacorr =

0BBB@�2syst;1 �syst;1�syst;2 �syst;1�syst;3

�syst;2�syst;1 �2syst;2 �syst;2�syst;3

�syst;3�syst;1 �syst;3�syst;2 �2syst;3

1CCCA =

0BBB@0:068 0:078 0:901

0:078 0:090 0:105

0:091 0:105 0:123

1CCCA

107

The uncorrelated theory error is also diagonal:

Etheoryuncorr

0BBB@0:010 0 0

0 0:014 0

0 0 0:020

1CCCAThe sum of all three matrices is the covariance matrix C:

C = Edatauncorr+ Edata

corr +Etheoryuncorr =

0BBB@0:103 0:078 0:091

0:078 0:137 0:105

0:091 0:105 0:186

1CCCAThe formula for the �2 between the data and theory is

�2 =3Xi j

(Di � Ti) 1

Cij(Dj � Tj)

To calculate the term in the sum we need the inverse of C:

C�1 =

0BBB@20:18 �6:94 �5:95�6:94 15:28 �5:22�5:95 �5:22 11:22

1CCCAThe (row=1,column=2) term is

term1;2 = (D1 � T1)� C�112 � (D2 � T2)

= (5:20� 5:00)� �6:94� (5:9� 6)

= 0:139

This is repeated for all other terms in the 3 � 3 matrix. The number of calculations

can be reduced to almost half by using the fact that the inverse covariance matrix is

symmetric about the diagonal. The sum of all nine terms in this example is 1.216, so the

108

reduced �2 is 1.216/3 = 0.41. The size of the covariance matrix is equal to the number

of data points in the Data-Theory comparison. For the comparison of the cross section

ratio to theory the covariance matrix is 28� 28.